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AppliedMath
, Volume 3 (2) – Apr 4, 2023

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Article Radial Based Approximations for Arcsine, Arccosine, Arctangent and Applications Roy M. Howard School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth 6845, Australia; r.howard@curtin.edu.au Abstract: Based on the geometry of a radial function, a sequence of approximations for arcsine, arcco- sine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is signiﬁcantly better than Taylor series approximations for arguments approaching one. The established approxima- tions can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of 10 , and lower, can be deﬁned. Applications of the approximations include: ﬁrst, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with signiﬁcantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approxima- tions to arccosine and arcsine, to even order powers, with relative errors that are signiﬁcantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals. Keywords: arcsine; arccosine; arctangent; two point spline approximation; upper and lower bounded functions; Newton-Raphson MSC: 26A09; 26A18; 26D05; 41A15 1. Introduction Citation: Howard, R.M. Radial Based The elementary trigonometric functions are fundamental to many areas of mathematics Approximations for Arcsine, with, for example, Fourier theory being widely used and ﬁnding widespread applications. Arccosine, Arctangent and The formulation of trigonometric results was pre-dated by interest in the geometry of Applications. AppliedMath 2023, 3, triangles and this occurs well in antiquity, e.g., [1]. The fundamental functions of sine and 343–394. https://doi.org/10.3390/ cosine have a geometric basis and are naturally associated with an angle from the positive appliedmath3020019 horizontal axis to a point on the unit circle. From angle addition and difference identities for sine and cosine, the derivatives of these functions can be deﬁned and, subsequently, Received: 30 September 2022 Taylor series approximations for sine and cosine can be established. Such approximations Revised: 12 December 2022 Accepted: 15 December 2022 have reasonable convergence with a ninth order expansion having a relative error bound Published: 4 April 2023 of 3.54 10 for the interval [0, p/2]. Naturally, many other approximations have been developed, e.g., [2–4]. The inverse trigonometric functions of arcsine, arccosine and arctangent are naturally of interest and ﬁnd widespread use for both the general complex case and the real case. Copyright: © 2023 by the author. The arctangent function, for example, is found in the solution of the sine-Gordon partial Licensee MDPI, Basel, Switzerland. differential equation for the case of soliton wave propagation, e.g., [5]. In statistical analysis This article is an open access article the arcsine distribution is widely used and the arctangent function is the basis of a wide distributed under the terms and class of distributions, e.g., [6]. The graphs of sine, cosine, arcsine and arccosine are shown conditions of the Creative Commons in Figure 1. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). AppliedMath 2023, 3, 343–394. https://doi.org/10.3390/appliedmath3020019 https://www.mdpi.com/journal/appliedmath AppliedMath 2022, 2, FOR PEER REVIEW 2 problem with respect to finding approximations is that both arcsine and arccosine have undefined derivatives at the point one. An overview of established approximations for arcsine and arctangent is provided in Section 2. In this paper, a geometric approach based on a radial function, whose derivatives are well defined at the point one, is used to estab- lish new approximations for arccosine, arcsine and arctangent. The approximations for arccosine and arcsine are sharp (zero relative error) at the points zero and one and have a defined relative error bound over the interval [0,1]. Convergence of the approximations is proved and the convergence is significantly better, for arguments approaching one, than Taylor series approximations. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and AppliedMath 2023, 3 344 with relative error bounds of the order of 10 , and lower, can be defined. 1 1 Figure 1. Graph of y = f (x) = sin(x), x = f (y) = asin(y), y = g(x) = cos(x) and x = g (y) = ( ) ( ) ( ) ( ) Figure 1. Graph of 𝑦= 𝑓 𝑥 =sin(𝑥) , 𝑥= 𝑓 𝑦 =asin(𝑦) , 𝑦= 𝑔 𝑥 =cos(𝑥) and 𝑥= 𝑔 𝑦 = acos(y) for 0 x , 0 y 1. Arcsine and arccosine are, respectively, written as asin and acos. acos(𝑦) for 0≤𝑥 ≤ , 0 ≤𝑦 ≤1 . Arcsine and arccosine are, respectively, written as asin and acos. Taylor series expansions for arcsine and arccosine, unlike those for sine and cosine, Applications for the established approximations are detailed and these include: First, have relatively poor convergence properties over the interval [0, 1] and a potential problem approximations for arcsine, arccosine and arctangent to achieve a set relative error bound. with respect to ﬁnding approximations is that both arcsine and arccosine have undeﬁned Second, upper and lower bounded approximations, of arbitrary accuracy, for arcsine, arc- derivatives at the point one. An overview of established approximations for arcsine and cosine and arctangent. Third, approximations to arccosine and arcsine, of even order pow- arctangent is provided in Section 2. In this paper, a geometric approach based on a radial ers, which have significantly lower relative error bounds than published approximations. function, whose derivatives are well deﬁned at the point one, is used to establish new Fourth, approximations for the inverse tangent integral function with significantly lower approximations for arccosine, arcsine and arctangent. The approximations for arccosine relative error bounds, over the interval [0,∞), than established Taylor series based ap- and arcsine are sharp (zero relative error) at the points zero and one and have a deﬁned proximations. Fifth, examples of approximations for unknown integrals. relative error bound over the interval [0, 1]. Convergence of the approximations is proved and the convergence is signiﬁcantly better, for arguments approaching one, than Taylor 1.1. Fundamental Relationships series approximations. The established approximations can be utilized as the basis for For the real case the following relationships hold: Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of 10 , and lower, can be deﬁned. 𝑎𝑠𝑖𝑛 (−𝑦 ) =−𝑖𝑛𝑠𝑎 (𝑦 ),𝑎𝑐𝑜𝑠 (−𝑦 ) =𝜋 − 𝑐𝑎𝑜𝑠 (𝑦 ), 𝑦 ∈ [0,1] (1) Applications for the established approximations are detailed and these include: First, ( ) ( ) [ ) 𝑎𝑡𝑎𝑛 −𝑦 = −𝑎𝑡𝑎𝑛 𝑦 ,𝑦∈ 0,∞ approximations for arcsine, arccosine and arctangent to achieve a set relative error bound. Thus, it is sufficient to detail approximations over the interval [0,1] for arcsine and arc- Second, upper and lower bounded approximations, of arbitrary accuracy, for arcsine, ar- cosine and approximations over the positive real line for arctangent. ccosine and arctangent. Third, approximations to arccosine and arcsine, of even order Fundamental relationships for arcsine, arccosine and arctangent, e.g., [7] (1.623, powers, which have signiﬁcantly lower relative error bounds than published approxima- 1.624, p. 57) are: tions. Fourth, approximations for the inverse tangent integral function with signiﬁcantly lower relative error bounds, over the interval [0,¥), than established Taylor series based (𝑦) = − 𝑐𝑎𝑜𝑠(𝑦), 𝑖𝑛( 𝑠𝑎 𝑦) = 𝑐𝑎𝑜𝑠 1−𝑦 , approximations. Fifth, 2 examples of approximations for unknown integrals. (2) 𝑎𝑐𝑜𝑠(𝑦) = − 𝑖𝑛( 𝑠𝑎 𝑦), 𝑐𝑎𝑜𝑠(𝑦) = 𝑖𝑛𝑠𝑎 1−𝑦 , 0 ≤ 𝑦 ≤ 1, 1.1. Fundamental Relationships 2 For the real case the following relationships hold: 𝑦 1−𝑦 (𝑦) = , 𝑐𝑎𝑜𝑠(𝑦) = 𝑡𝑎𝑛𝑎 , 0 ≤𝑦 ≤1 (3) 1− 𝑦 asin(y) = asin(y), acos(y) = p acos(y), y 2 [0, 1] (1) atan(y) = atan(y), y 2 [0,¥) Thus, it is sufﬁcient to detail approximations over the interval [0, 1] for arcsine and arccosine and approximations over the positive real line for arctangent. Fundamental relationships for arcsine, arccosine and arctangent, e.g., [7] (1.623, 1.624, p. 57) are: h i asin(y) = acos(y), asin(y) = acos 1 y , (2) h i acos(y) = asin(y), acos(y) = asin 1 y , 0 y 1, " # " # y 1 y asin(y) = atan p , acos(y) = atan , 0 y 1 (3) 1 y 𝑎𝑡𝑎𝑛 𝑎𝑠𝑖𝑛 𝑎𝑠𝑖𝑛 AppliedMath 2023, 3 345 " # " # y p y atan(y) = asin p = acos p 2 2 1 + y 1 + y (4) " # " # 1 p 1 p p atan(y) = acos = asin , 0 y ¥. 2 2 1 + y 1 + y These relationships imply, for example, that approximations for arcsine and arctangent follow from an approximation to arccosine and approximations for arcsine and arccosine follow from an approximation to arctangent. 1.2. Notation For an arbitrary function f , deﬁned over the interval [a, b], an approximating function f has a relative error, at a point x , deﬁned according to re(x ) = 1 f (x )/ f (x ). The A 1 1 A 1 1 relative error bound for the approximating function, over the interval [a, b], is deﬁned according to re = maxfjre(x )j : x 2 [a, b]g (5) B 1 1 (k) The notation f is used for the kth derivative of a function. In equations, arcsine, arccosine and arctangent are abbreviated, respectively, as asin, acos and atan. Mathematica has been used to facilitate analysis and to obtain numerical results. In general, the relative error results associated with approximations to arcsine, arccosine and arctangent have been obtained by sampling speciﬁed intervals, in either a linear or logarithmic manner, as appropriate, with 1000 points. 1.3. Paper Structure A review of published approximations for arcsine and arctangent is provided in Section 2. In Section 3, the geometry, and analysis, of the radial function that underpins the proposed approximations for arccosine, arcsine and arctangent, is detailed. In Section 4, convergence of the approximations is detailed. In Section 5, the antisymmetric nature of the arctangent function is utilized to establish spline based approximations for this function. In Section 6, iteration, based on the proposed approximations, is utilized to detail approximations with quadratic convergence. Applications of the proposed approximations are detailed in Section 7 and conclusions are stated in Section 8. 2. Published Approximations for Arcsine and Arctangent The Taylor series expansions for arcsine and arctangent, respectively, are, e.g., [8] (eqns. 4.24.1, 4.24.3, 4.24.4, p. 121) h i k1 2k+1 3 5 7 9 Õ 2i + 1 y y 3y 5y 35y i=0 asin(y) = y + + + + + = y + 6 40 112 1152 (2k + 1) 2i k=1 Õ i=1 (6) 2k+1 (2k)!y = , 0 y < 1 2k k=0 2 (2k + 1)(k!) 3 5 7 9 2k+1 > y y y y (1) y > y + + + . . . + + . . . , 0 y < 1 3 5 7 9 2k + 1 atan(y) = (7) k+1 p 1 1 1 1 1 (1) + + + . . . + + . . . , y 1 3 5 7 9 (2k+1) 2 y 3y 5y 7y 9y (2k + 1)y For a set order, the relative error in a Taylor series approximation for arcsine increases sharply as y ! 1 (see Figure 2). AppliedMath 2023, 3 346 2.1. Approximation Form for Arcsine The nature of arcsine is such that it has a rate of change of 1 at the origin and an inﬁnite rate of change at the point one which complicates ﬁnding suitable approxima- tions. An approximation form that has potential is 1 1 y, whose rate of change is 1/2 1 y, with the rate of change being 1/2 at the origin. As a starting point, consider the approximation form h i s(y) = a 1 1 y + a y + a y (8) 0 1 2 The three coefﬁcients can be chosen to satisfy the constraints consistent with a sharp ap- (1) (1) proximation at the points zero and one: s(0) = 0, s(1) = p/2, s (0) = 1 and s (1) = ¥. The constraints imply a = 1 a /2, a = p/2 a /2 1, with a being arbitrary. For the 1 0 2 0 0 case of a = p/2, the approximation is h i h i h i p p p s (y) = 1 1 y + 1 y 1 y (9) 2 4 4 which has a relative error bound, for the interval [0, 1], of 2.66 10 . 2.1.1. Optimized Coefﬁcients The coefﬁcient a can be optimized consistent with minimizing the relative error bound over the interval [0, 1]. The optimum coefﬁcient of a = p/2 1306/10, 000 leads to the approximation s (y) = a 1 1 y + a y + a y , 2 0 1 2 (10) p 1306 10, 653 p p 9347 a = , a = , a = , 0 1 2 2 10, 000 10, 000 4 4 10, 000 which has a relative error bound, for the interval [0, 1], of 3.62 10 . 2.1.2. Padè Approximants Given a suitable approximation form, Padè approximants can be utilized to find approx- imations with lower relative error bounds. For example, the form p/2 1 y p (y), n,m where p is an approximant of order n, m, can be utilized. n,m 2.2. Published Approximations The arcsine case is considered as related approximations for arccosine and arctangent follow from Equations (2) and (4). The following approximations are indicative of published approximations. First, the approximation py s (y) = h i , y 2 [0, 1] (11) 3 p 2 y + 1 y arises from the simple approximation for arctangent, e.g., [9] (eqn. 5), of py atan(y) , y 2 [0,¥) (12) 2(1 + y) The maximum error in this approximation has a magnitude of 0.0711, but the relative error bound is 0.571, which occurs as y approaches zero. AppliedMath 2023, 3 347 asin(y) yasin(y) Second, a Taylor series expansion for p , e.g., [10] (eqn. 4) or p , e.g. [11], 2 2 1 y 1 y can be used. The latter yields the nth order approximation: 2k1 2k1 2 (k!) y s (y) = 1 y 4,n k(2k)! k=1 (13) 3 5 7 9 2y 8y 16y 128y = 1 y y + + + + + 3 15 35 315 Consistent with a Taylor series, the relative error is low for jyj 1 but, for a set order, becomes increasingly large as y ! 1 . Third, the following approximations are stated in [12] (eqns. 1.5 and 3.7): " # 5 1 y 80y 1 + py p " # s (y) = , s (y) = (14) 5 6 p 2 2 2 + 1 y 2y 16 1 y 57 1 + 19 19 The ﬁrst approximation is part of the Shafer-Fink inequality (e.g., [13]) is not sharp at the origin and has a relative error bound, for the interval [0, 1], of 4.72 10 . The second approximation is not sharp at y = 1 but has a relative error bound for the interval [0, 1], of 1.38 10 . Fourth, the following approximation is detailed in [14] (eqn. 4.4.46, p. 81): h i 2 7 s (y) = 1 y a + a y + a y + . . . + a y (15) 7 0 1 2 7 where a = , a = 0.2145988016, a = 0.0889789874, 0 1 2 (16) a = 0.0501743046, a = 0.0308918810, a = 0.0170881256, 3 4 5 a = 0.0066700901, a = 0.0012624911. 6 7 The relative error bound is 3.04 10 which occurs at the origin. Fifth, [15] (Section 6.4), provides a basis for determining approximations for arcsine, arccosine and arctangent of arbitrary accuracy. Explicit formulas and results are detailed in Appendix A. For example, the following approximation for arcsine (as deﬁned by h i c 1 y —see Equation (A13)) is 2,2 p p 2 3 121 2 2 1 + 1 y 1 + 1 y 4 5 s (y) = 1 p + 121 2 p p 2 3 p p (17) 2 2 1 + 1 y 38 2 1 + 1 y 178 74 1 y 4 5 + + 5/2 15 15 5 p p 2 + 2 1 + 1 y and has a relative error bound of 1.71 10 that occurs at y = 1. AppliedMath 2022, 2, FOR PEER REVIEW 5 Third, the following approximations are stated in [12] (eqns. 1.5 and 3.7): 5 1−𝑦 80𝑦 1 + 𝜋𝑦 𝑠 (𝑦) = , 𝑠 (𝑦) = (14) 2+ 1−𝑦 2𝑦 16 1−𝑦 57 1 − + 19 19 The first approximation is part of the Shafer-Fink inequality (e.g., [13]) is not sharp at the −2 origin and has a relative error bound, for the interval [0,1], of 4.72 × 10 . The second ap- proximation is not sharp at 𝑦= 1 but has a relative error bound for the interval [0,1], of −3 1.38 × 10 . Fourth, the following approximation is detailed in [14] (eqn. 4.4.46, p. 81): [ ] 𝑠 (𝑦) = − 1−𝑦 ⋅ 𝛼 +𝛼 𝑦+ 𝛼 𝑦 +⋯ +𝛼 𝑦 (15) where 𝛼 = , 𝛼 = −0.2145988016, 𝛼 = 0.0889789874, (16) 𝛼 = −0.0501743046, 𝛼 = 0.0308918810, 𝛼 = −0.0170881256, 𝛼 = 0.0066700901, 𝛼 = −0.0012624911. −6 The relative error bound is 3.04 × 10 which occurs at the origin. Fifth, [15] (Section 6.4), provides a basis for determining approximations for arcsine, arccosine and arctangent of arbitrary accuracy. Explicit formulas and results are detailed in Appendix A. For example, the following approximation for arcsine (as defined by 𝑐 1− 𝑦 —see Equation (A13)) is 121 2− √2 1+ 1− 𝑦 ⎡ ⎤ 1+ 1− 𝑦 ⎢ ⎥ 𝑠 (𝑦 ) = ⋅ 1− + ⎢ 121√2 ⎥ ⎣ ⎦ (17) 2− 2 1+ 1−𝑦 ⎡ ⎤ 38 2 1+ 1− 𝑦 178 74 1− 𝑦 ⎢ ⎥ ⋅ + + 15 15 5 ⎢ ⎥ 2 + 2 1+ 1− 𝑦 ⎣ ⎦ AppliedMath 2023, 3 348 −5 and has a relative error bound of 1.71 × 10 that occurs at 𝑦= 1 . Comparison of Approximations Comparison of Approximations The graphs of the relative errors associated with the above approximations are shown The graphs of the relative errors associated with the above approximations are shown in Figure 2. in Figure 2. AppliedMath 2022, 2, FOR PEER REVIEW 6 Figure 2. Graphs of the magnitude of the relative error in approximations to arcsine as deﬁned in the Figure 2. Graphs of the magnitude of the relative error in approximations to arcsine as defined in text. Taylor series approximations, of orders 3, 7, 11, 15, 19, 23, are shown dotted. the text. Taylor series approximations, of orders 3, 7, 11, 15, 19, 23, are shown dotted. 3. Radial Based Two Point Spline Approximation for Arccosine Squared 3. Radial Based Two Point Spline Approximation for Arccosine Squared Consider the geometry, as illustrated in Figure 3, associated with arcsine and arccosine Consider the geometry, as illustrated in Figure 3, associated with arcsine and arc- and which underpins the four radial functions deﬁned according to cosine and which underpins the four radial functions defined according to h i 2 2 2 r (y) = y + asin(y) = y + acos(y) , (18) (18) 𝑟 (𝑦) = 𝑦 + −𝑎𝑖𝑛𝑠(𝑦) =𝑦 +𝑎𝑐𝑜𝑠(𝑦) , h i 𝜋 2 2 2 2 (19) 𝑟 (𝑦) = (1 − 𝑦) + −𝑎𝑖𝑛𝑠(𝑦) =(1 − 𝑦) +𝑎𝑐𝑜𝑠(𝑦) , r (y) = (1 y) + asin(y) = (1 y) + acos(y) , (19) 1 2 𝑟 (𝑦) = 𝑦 +𝑎𝑖𝑛𝑠(𝑦) , 𝑟 (𝑦) = (1 − 𝑦) +𝑎𝑖𝑛𝑠(𝑦) , 𝑦 ∈ [0,1]. (20) 2 2 2 2 2 2 r (y) = y + asin(y) , r (y) = (1 y) + asin(y) , y 2 [0, 1]. (20) 2 3 Figure 3. Illustration of four radial functions associated with arcsine and arccosine. Figure 3. Illustration of four radial functions associated with arcsine and arccosine. 2 2 The graphs of The gr these fun aphs o ctions f these are functions are shown in Figur show e 4.n in TheFig functions ure 4. The func r and r tion have s 𝑟 and 𝑟 have 2 3 undeﬁned derivatives at the point y = 1, which does not facilitate function approximation. undefined derivatives at the point 𝑦= 1 , which does not facilitate function approxima- 2 2 The function r is smoother than r and can be utilized as a basis for approximation. If there tion. The function 𝑟 is smoother than 𝑟 and can be utilized as a basis for approxima- exists an nth ortion. I der appr f ther oximation, e exists an f , to 𝑛𝑡ℎ r , then order theapprox relationships imation a,c o𝑓 s(, to y) 𝑟 , f th (yen the ) y , relationships n n h i (𝑦) ≈ 𝑓 (𝑦) − 𝑦 , (𝑦) ≈ −𝑎𝑐𝑜𝑠( 𝑦) and 𝑎𝑡𝑎𝑛( 𝑦) = 𝑎𝑐𝑜𝑠1/ 1+𝑦 can be asin(y) acos(y) and atan(y) = acos 1/ 1 + y can be utilized to establish approx- utilized to establish approximations for arccosine, arcsine and arctangent. imations for arccosine, arcsine and arctangent. Figure 4. Graph of 𝑟 (𝑦 ), 𝑟 (𝑦 ),𝑟 (𝑦) 𝑎𝑑𝑛 𝑟 (𝑦 ). 3.1. Approximations for Radial Function The two point spline approximation detailed in [15] (eqn. 40), and the alternative form given in [16] (eqn. 70) can be utilized to establish convergent approximations to the radial function 𝑟 defined by Equation (18). Theorem 1. Two Point Spline Approximations for Radial Function. The 𝑛𝑡ℎ order two point spline approximation to the radial function 𝑟 , based on the points zero and one, is 𝑎𝑠𝑖𝑛 𝑎𝑐𝑜𝑠 AppliedMath 2022, 2, FOR PEER REVIEW 6 Figure 2. Graphs of the magnitude of the relative error in approximations to arcsine as defined in the text. Taylor series approximations, of orders 3, 7, 11, 15, 19, 23, are shown dotted. 3. Radial Based Two Point Spline Approximation for Arccosine Squared Consider the geometry, as illustrated in Figure 3, associated with arcsine and arc- cosine and which underpins the four radial functions defined according to 𝑟 (𝑦) = 𝑦 + −𝑎𝑖𝑛𝑠(𝑦) =𝑦 +𝑎𝑐𝑜𝑠(𝑦) , (18) (19) 𝑟 (𝑦) = (1 − 𝑦) + −𝑎𝑖𝑛𝑠(𝑦) =(1 − 𝑦) +𝑎𝑐𝑜𝑠(𝑦) , [ ] (20) 𝑟 (𝑦) = 𝑦 +𝑎𝑖𝑛𝑠(𝑦) , 𝑟 (𝑦) = (1 − 𝑦) +𝑎𝑖𝑛𝑠(𝑦) , 𝑦 ∈ 0,1 . Figure 3. Illustration of four radial functions associated with arcsine and arccosine. The graphs of these functions are shown in Figure 4. The functions 𝑟 and 𝑟 have undefined derivatives at the point 𝑦= 1 , which does not facilitate function approxima- tion. The function 𝑟 is smoother than 𝑟 and can be utilized as a basis for approxima- tion. If there exists an 𝑛𝑡ℎ order approximation, 𝑓 , to 𝑟 , then the relationships AppliedMath 2023, 3 349 (𝑦) ≈ 𝑓 (𝑦) − 𝑦 , (𝑦) ≈ −𝑎𝑐𝑜𝑠( 𝑦) and 𝑎𝑡𝑎𝑛( 𝑦) = 𝑎𝑐𝑜𝑠1/ 1+𝑦 can be utilized to establish approximations for arccosine, arcsine and arctangent. 2 2 2 2 Figure 4. Graph of r (y), r (y), r (y) and r (y). Figure 4. Graph of 𝑟 (𝑦 ), 𝑟 (𝑦 ),𝑟 (𝑦) 𝑎𝑑𝑛 𝑟 (𝑦 ). 1 2 3 3.1. Approximations for Radial Function 3.1. Approximations for Radial Function The two point spline approximation detailed in [15] (eqn. 40), and the alternative form The two point spline approximation detailed in [15] (eqn. 40), and the alternative given in [16] (eqn. 70) can be utilized to establish convergent approximations to the radial form given in [16] (eqn. 70) can be utilized to establish convergent approximations to the function r deﬁned by Equation (18). radial function 𝑟 defined by Equation (18). Theorem 1. Two Point Spline Approximations for Radial Function. Theorem 1. Two Point Spline Approximations for Radial Function. The nth order two point spline approximation to the radial function r , based on the points The 𝑛𝑡ℎ order two point spline approximation to the radial function 𝑟 , based on the points zero and one, is zero and one, is 2n+1 f (y) = C y , n 2 0, 1, 2, . . . (21) f g n å n,k k=0 where the coefﬁcients C are deﬁned according to: n,k kr (k) (1) (n + 1)! f (0) a = , 0 k n > å n,r > (n + 1 + r k)!(k r)! k! r=0 kr (1) n + 1 ! ( ) C = a + (22) n,k n,r (n + 1 + r k)!(k r)! > r=kn1 kn1 > (1) r! b , n + 1 k 2n + 1 : n,r (r + n + 1 k)!(k n 1)! r=kn1 Here f (y) = r (y) and ru r r (ru) (ru) f (0) (n + u)! (1) f (1) (n + u)! a = , b = (23) n,r å n,r å (r u)! u!n! (r u)! u!n! u=0 u=0 r 2 f0, 1, . . . , ng. The derivative values of f , at the points zero and one, are deﬁned according to 2 (1) (3) (2) (4) f (0) = p /4, f (0) = f (0) = p, f (0) = 4, f (0) = 8, (24) (k) (k2) f (0) = (k 2) f (0), k 2 5, 6, 7, . . . f g (1) (2) (3) f (1) = 1, f (1) = 0, f (1) = 8/3, f (1) = 8/15, k 2 (25) (1) (k 1) (k) (k1) f (1) = f (1) , k 2 f4, 5, 6, . . .g. 2k 1 Proof. The proofs for these results are detailed in Appendix F. 𝑎𝑠𝑖𝑛 𝑎𝑐𝑜𝑠 AppliedMath 2023, 3 350 3.1.1. Notes on Coefﬁcients Explicit expressions for the coefﬁcients C , n 2 f0, 1, . . . , 6g, k 2 f0, 1, . . . , 2n + 1g, n,k are tabulated in Table A1 (Appendix B). (k) (k) (k2) As C = f (0)/k!, k 2 0, 1, . . . , n , and f (0) = (k 2) f (0), k 2 5, 6, . . . , f g f g n,k it follows that 2 2 (k) (k2) f (0) (k 2) f (0) (1 2/k) n 2 f5, 6, . . .g C = = = C , (26) n,k n,k2 k! k(k 1) (k 2)! 1 1/k k 2 f5, 6, . . . , ng (4) (2) 2 (3) As C = f 0 /4! = 1/3, C = f 0 /2 = 2, C = p /4, C = f 0 /3! = ( ) ( ) ( ) n,4 n,2 n,0 n,3 (1) p/6, and C = f (0) = p, it is the case that C < C for k 2 f2, 3, . . . , ng, n,1 n,k n,k2 n 2. Hence, for n ﬁxed, n 3, the magnitudes of both even and odd order coefﬁcients monotonically decrease as k increases and for k 2 f3, 4, . . . , ng. 3.1.2. Explicit Approximations Explicit approximations for r , of orders zero and one, are: 2 2 p p f (y) = + y 1 (27) 4 4 2 2 2 p 3p p 2 3 f y = py + 3 + 2p y + 2 p + y (28) ( ) 4 4 2 Higher order approximations, up to order six, are detailed in Appendix B along with the relevant coefﬁcients C , k 2 f0, 1, . . . , 2n + 1g (see Table A1). n,k 3.1.3. Approximations for Arccosine, Arcsine and Arctangent With the deﬁnition of 1, n = 0, k = 2 c = C 1, k = 2, n 2 f1, 2, . . .g (29) n,k n,k C , k 2 0, 1, 3, . . . , 2n + 1 , n 2 0, 1, . . . f g f g n,k the approximations, as stated in Corollary 1, follow. Corollary 1. Approximations for Arccosine, Arcsine and Arctangent. The approximations for arccosine, arcsine and arctangent arising from the approxima- tions speciﬁed in Theorem 1 are: v v u u 2n+1 2n+1 u u k/2 t t k 2 acos(y) c y = c y , acos y c y = c 1 y , (30) ( ) ( ) ( ) ( ) å n,k n å n,k k=0 k=0 v v u u 2n+1 2n+1 u u k/2 t A t k 2 asin(y) s (y) = c y , asin(y) s (y) = c (1 y ) , (31) å n,k n å n,k k=0 k=0 v v u u 2n+1 2n+1 k u u c p c y n,k n,k t A t atan(y) t (y) = , atan(y) t (y) = , (32) å n å k/2 k/2 2 2 (1 + y ) (1 + y ) k=0 k=0 for n 2 1, 2, 3, . . . . The superscript A denotes alternative approximation forms. For the f g case of n= 0, the upper limit of the summations is 2 rather than 1. AppliedMath 2023, 3 351 2 2 Proof. These results follow directly from the definition acos(y) = r y y (Equation (18)), ( ) and the approximations f (y) = r (y) detailed in Theorem 1, leading to u u q maxf2,2n+1g 2n+1 u u t t 2 k 2 k acos(y) f (y) y = C y y = c y (33) å n,k å n,k k=0 k=0 The approximations for the other results arise from the fundamental relationships detailed in Equations (2)–(4), and according to " # p 1 s (y) = c (y), t (y) = c p , n n n n 1 + y (34) " # h i h i p p A A A 2 2 s (y) = c 1 y ,c (y) = s 1 y ,t (y) = s p . n n n n n n 1 + y 3.1.4. Explicit Approximations for Arccosine, Arcsine and Arctangent Explicit approximations for arccosine, of orders zero, one and two, are: 2 2 p p c (y) = + y 1 y (35) 4 4 2 3 c (y) = py + c y + c y (36) 1 1,2 1,3 2 3 4 5 y = py + y + c y + c y + c y (37) c ( ) 2 2,3 2,4 2,5 Approximations, of orders three to six, are detailed in Appendix C. Explicit approxi- mations for arcsine, of orders zero to six, can then be speciﬁed by utilizing the rela- h i tionships s y = p/2 c y and s y = c 1 y , i 2 0, 1, . . . , 6 . Explicit ap- ( ) ( ) ( ) f g i i i h i proximations for arctangent follow from the relationships t y = c 1/ 1 + y and ( ) i i h i A 2 t (y) = s y/ 1 + y , i 2 f0, 1, . . . , 6g. For example, the second order approximation for arctangent is p p 1 c c c 2,3 2,4 2,5 t (y) = p + + + + (38) 2 3/2 2 5/2 2 2 2 4 1 + y 1 + y [1 + y ] [1 + y ] [1 + y ] 3.1.5. Relative Error Bounds for Arcsine, Arccosine and Arctangent The relative error bounds for the approximations to r , arcsine, arccosine and arctan- gent, arising from the approximations stated in Theorem 1 and Corollary 1 are detailed in Table 1. The relative errors in the approximations, of orders one to ﬁve, for arcsine, arccosine and arctangent are shown in Figures 5 and 6. For example, the relative error bound associated with the fourth, s (y), and sixth, s (y), order approximations to arcsine, 4 6 6 8 respectively, are 2.49 10 and 2.28 10 . AppliedMath 2022, 2, FOR PEER REVIEW 9 AppliedMath 2022, 2, FOR PEER REVIEW 9 (37) 𝑐 (𝑦) = −𝜋𝑦 + 𝑦 +𝑐 𝑦 +𝑐 𝑦 +𝑐 𝑦 , , , (37) 𝑐 (𝑦) = −𝜋𝑦 + 𝑦 +𝑐 𝑦 +𝑐 𝑦 +𝑐 𝑦 4 , , , AppliedMath 2023, 3 352 Approximations, of orders three to six, are detailed in Appendix C. Explicit approxima- Approximations, of orders three to six, are detailed in Appendix C. Explicit approxima- tions for arcsine, of orders zero to six, can then be specified by utilizing the relationships tions for arcsine, of orders zero to six, can then be specified by utilizing the relationships 𝑠 ( T 𝑦 able ) =𝜋1.2 ⁄Relative −𝑐 (𝑦 ) er and ror bounds 𝑠 (𝑦 ) =𝑐 for appr 1−𝑦 oximations , 𝑖∈ to {0,1, r , … ar,csine, 6}. Expl arccosine icit approxim and arctangent. ations for The ( ) ⁄ ( ) ( ) { } 𝑠 𝑦 =𝜋 2 −𝑐 𝑦 and 𝑠 𝑦 =𝑐 1−𝑦 , 𝑖∈ 0,1, … ,6 . Explicit approximations for interval [0, 1] is assumed for r , arcsine and arccosine whilst the interval [0, ¥) is assumed for ( ) ⁄ ( ) arctangent follow from the relationships 𝑡 𝑦 =𝑐 1 1+ 𝑦 and 𝑡 𝑦 = ( ) ⁄ ( ) arctangent follow from the relationships 𝑡 𝑦 =𝑐 1 1+ 𝑦 and 𝑡 𝑦 = arctangent. ⁄ { } 𝑠 1+𝑦 , 𝑖∈ 0,1, … ,6 . For example, the second order approximation for arctangent ⁄ { } 𝑠 1+𝑦 , 𝑖∈ 0,1, … ,6 . For example, the second order approximation for arctangent is is Order of Relative Error Relative Error Bound: Relative Error Bound: 2 A A A Approx. Bound: r s (y), c (y), t (y) s (y), c (y), t (y) n n n n n n 𝜋 𝜋 1 𝑐 𝑐 𝑐 , , , 𝜋 𝜋 1 𝑐 𝑐 𝑐 ( ) 1 , 1 , , 1 𝑡 𝑦 = − + + + + (38) 0 3.01 10 5.33 10 3.17 10 ⁄ ⁄ ( ) 𝑡 𝑦 = − + + + + (38) 4 1+𝑦 [ ] [1+ 𝑦 ] [ ] 1+𝑦 1+𝑦 ⁄ ⁄ 1+𝑦 3 3 3 4 1+𝑦 [ ] [1+ 𝑦 ] [ ] 1+𝑦 1+𝑦 4.22 1+𝑦 10 5.79 10 2.92 10 4 4 4 2 2.77 10 3.64 10 1.81 10 5 5 5 3 2.20 10 2.84 10 1.42 10 3.1.5. Relative Error Bounds for Arcsine, Arccosine and Arctangent 6 6 6 3.1.5. Relat4 ive Error Bounds for Arcsine, Arccosine and Arctangent 1.95 10 2.49 10 1.24 10 7 7 7 The relative error bounds for the approximations to 𝑟 , arcsine, arccosine and arctan- 5 1.84 10 2.33 10 1.16 10 The relative error bounds for the approximations to 𝑟 , arcsine, arccosine and arctan- 8 8 8 6 1.81 10 2.28 10 1.14 10 gent, arising from the approximations stated in Theorem 1 and Corollary 1 are detailed in gent, arising from the approximations stated in Theorem 1 and Corollary 1 are detailed in 10 10 10 1.92 10 2.41 10 1.20 10 Table 1. The relative errors in the approximations, of orders one to five, for arcsine, arc- Table 1. The relative errors in the approximations, of orders one to five, for arcsine, arc- 12 12 12 10 2.21 10 2.76 10 1.38 10 cosine and arctangent are shown in Figures 5 and 6. For example, the relative error bound cosine and arctangent are shown in 14 Figures 5 and 6. For exam 14 ple, the relative error b14 ound 2.68 10 3.34 10 1.66 10 associated with the fourth, 𝑠 (𝑦 ), and sixth, 𝑠 (𝑦 ), order approximations to arcsine, re- 18 18 18 associated with the fourth, 𝑠 (𝑦 ), and sixth, 𝑠 (𝑦 ), order approximations to arcsine, re- 16 4.35 10 5.41 10 2.70 10 spectively, are 2.49 × 10 and 2.28 × 10 . spectively, are 2.49 × 10 and 2.28 × 10 . Figure 5. Graph of the relative error in approximations, of orders 1 to 5, for arcsine and arccosine. Figure 5. Graph of the relative error in approximations, of orders 1 to 5, for arcsine and arccosine. Figure 5. Graph of the relative error in approximations, of orders 1 to 5, for arcsine and arccosine. 𝐴 𝐴 A A ( ) ( ) { } The dotted curves are for the approximations 𝑠 𝑦 and 𝑐 𝑦 , 𝑛∈ 1,2, … ,5 . The dotted curves are for the approximations 𝐴 s (y) and 𝐴 c (y), n 2 f1, 2, . . . , 5g. 𝑛 𝑛 n n ( ) ( ) { } The dotted curves are for the approximations 𝑠 𝑦 and 𝑐 𝑦 , 𝑛∈ 1,2, … ,5 . 𝑛 𝑛 Figure 6. Graph of the relative error in approximations, of orders 1 to 5, for arctangent. Figure 6. Graph of the relative error in approximations, of orders 1 to 5, for arctangent. Figure 6. Graph of the relative error in approximations, of orders 1 to 5, for arctangent. 3.2. Alternative Approximations I: Differentiation of Arccosine Squared Table 1. Relative error bounds for approximations to r , arcsine, arccosine and arctangent. The interval Table 1. Relative error bounds for approximations to r , arcsine, arccosine and arctangent. The interval Based on dif 2 ferentiation of the square of arccosine, alternative approximations for [0, 1] is assumed for r , arcsine and arccosine whilst the interval [0, ∞) is assumed for arctangent. [0, 1] is assumed for r , arcsine and arccosine whilst the interval [0, ∞) is assumed for arctangent. arccosine, arcsine and arctangent can be determined. Theorem 2. Alternative Approximations I: Differentiation of Arccosine Squared. Alternative approximations, of order n, n 2 f1, 2, . . .g, for arcsine and arccosine, over the interval [0, 1], and arctangent, over the interval [0, ¥), are: 𝑦 AppliedMath 2023, 3 353 2n 2n k/2 k A 2 asin(y) s (y) = 1 y d y , asin(y) s (y) = y d 1 y , (39) å n,k n å n,k k=0 k=0 2n 2n k/2 k A 2 acos(y) c (y) = 1 y d y , acos(y) c (y) = y d 1 y , (40) n å n,k å n,k k=0 k=0 2n 2n k d d y y p 1 n,k A n,k p p atan(y) t (y) = , atan(y) t (y) = , (41) å n å k/2 k/2 2 2 2 2 2 1 + y 1 + y (1 + y ) (1 + y ) k=0 k=0 (k + 1)c n,k+1 where d = , k 2 f0, 1, . . . , 2ng, with c being deﬁned by Equation (29). n,k n,k Proof. Consider the nth order approximation for arccosine, as deﬁned in Corollary 1: 2n+1 acos(y) p (y), where p (y) = c y ,n 2 f1, 2, . . .g. Assuming convergence, it n n n,k k=0 follows that acos(y) = p (y). Differentiation yields 2acos(y) (1) p = p (y), y 2 [0, 1), (42) 1 y which implies p 2n+1 1 y ic (1) n,i i1 acos(y) = p (y) c (y) = 1 y å y ¥ n 2 2 i=1 (43) p 2n = 1 y d y n,k k=0 after the index change of k = i 1 and where d = (k + 1)c /2. The approximation, n,k n,k+1 deﬁned by s , for arcsine follows from the relationship asin(y) = p/2 acos(y); the approximation for arctangent, deﬁned by t , follows according to " # 2n 1 y d n,k atan(y) = acos p t (y) = p (44) n å k/2 2 2 1 + y 1 + y (1 + y ) k=0 The alternative approximations follow according to " # q q A A 2 2 s (y) = c 1 y , c (y) = s 1 y , t (y) = s p . (45) n n n n n n 1 + y 3.2.1. Note The same approximations can be derived by considering the relationship atan(y) = 2atan(y)/ 1 + y which implies dy 1 + y d atan(y) = atan(y) (46) 2 dy AppliedMath 2023, 3 354 Use of the arctangent approximation, t y , specified in Corollary 1 leads to the approximation ( ) 2n+1 2p 1 + y ic y d n,i n,k atan(y) = y å å (k+1)/2 2 2 (1 + y ) i=1 k=0 (47) (1 + y ) (k + 1)c n,k+1 d = n,k after the change of index k = i 1. This result is consistent with t (y) stated in Theorem 2. 3.2.2. Explicit Approximations for Arcsine and Arctangent Approximations for arcsine, of orders one and two, are h i p p s (y) = 1 y + d y + d y 1 1,1 1,2 2 2 h i A 2 s (y) = y + d 1 y + d 1 y 1,1 1,2 (48) 2 2 3p 3p 3p d = 2 2p + , d = 3 + 1,1 1,2 4 2 4 h i p p 2 3 4 s (y) = 1 y y + d y + d y + d y 2 2,2 2,3 2,4 2 2 2 3 A 2 2 2 4 5 s (y) = y 1 y + d 1 y + d 1 y + d 1 y 2,2 2,3 (49) 2,4 2 2 2 15p 70 15p 40 15p 15p d = 8 9p + , d = + 16p , d = + 2,2 2,3 2,4 4 3 2 3 2 4 Approximations, of orders three and four, are detailed in Appendix D. As an example, the approximations for arctangent, of order two, are: " # y p 1 d d d 2,2 2,3 2,4 p p t (y) = + + + 2 3/2 2 2 2 2 2 2 1 + y 1 + y 1 + y (1 + y ) (1 + y ) (50) " # 2 3 4 p 1 p y d y d y d y 2,2 2,3 2,4 p p t (y) = + + + 2 3/2 2 2 2 2 2 2 2 1 + y 1 + y 1 + y (1 + y ) (1 + y ) 3.2.3. Results The relative error bounds associated with the approximations to arcsine, arccosine and arctangent, as speciﬁed by Theorem 2, are detailed in Table 2. The relative errors for arcsine, arccosine and arctangent are shown, respectively, in Figures 7–9. AppliedMath 2023, 3 355 Table 2. Relative error bounds, over the interval [0, 1] (arcsine and arccosine) and [0, ¥) (arctangent), associated with the approximations to arcsine, arccosine and arctangent as deﬁned in Theorem 2. A A A Order of Approx. Relative Error Bound: s ,c ,t Relative Error Bound: s ,c ,t n n n n n n 1 3 1 1.19 10 7.51 10 3 4 2 3.14 10 5.54 10 4 5 2.13 10 4.89 10 5 6 4 1.78 10 4.72 10 6 7 5 1.66 10 4.80 10 7 8 1.64 10 5.05 10 9 10 8 1.79 10 5.99 10 11 12 10 2.14 10 7.54 10 13 14 12 2.69 10 9.85 10 AppliedMath 2022, 2, FOR PEER REVIEW 12 AppliedMath 2022, 2, FOR PEER REVIEW 12 AppliedMath 2022, 2, FOR PEER REVIEW 12 17 17 16 4.71 10 1.81 10 Figure 7. Graph of the relative errors in the approximations, as defined in Theorem 2, to arcsine. Figure 7. Graph of the relative errors in the approximations, as defined in Theorem 2, to arcsine. Figure 7. Figure 7. Grap Graph h of the re of the rlativ elative e errors in errors in the theapprox approximations, imations, aas s define deﬁned d in in Theorem Theorem 2, 2, to toarcsine. arcsine. Figure 8. Graph of the relative errors in the approximations, as defined in Theorem 2, to arccosine. Figure 8. Graph of the relative errors in the approximations, as defined in Theorem 2, to arccosine. Figure 8. Figure 8. Grap Graph h of the re of the rlativ elative e errors in errors in the theapprox approximations, imations, aas s define deﬁned d in in Theorem Theorem 2, 2, to toarccosine arccosine. . Figure 9. Figure 9. Gr Graph aph of the relative errors in the approximat of the relative errors in the approximations, ions, as def as deﬁ ined in ned in Th Theor eorem 2, t em 2,oto arar ctangent. ctangent. Figure 9. Figure 9. G G rraph of the relative errors in the approximat aph of the relative errors in the approximations, as def ions, as def ii ned in ned in T T h h eorem 2, t eorem 2, t o o ar ar ctangent. ctangent. Table 2. Relative error bounds, over the interval [0, 1] (arcsine and arccosine) and [0, ∞) (arctangent), Table 2. Relative error bounds, over the interval [0, 1] (arcsine and arccosine) and [0, ∞) (arctangent), Table 2. Relative error bounds, over the interval [0, 1] (arcsine and arccosine) and [0, ∞) (arctangent), associated with the approximations to arcsine, arccosine and arctangent as defined in Theorem 2. associated with the approximations to arcsine, arccosine and arctangent as defined in Theorem 2. associated with the approximations to arcsine, arccosine and arctangent as defined in Theorem 2. 𝑨 𝑨 𝑨 𝑨 𝑨 𝑨 𝑨 𝑨 𝑨 Order of Approx. Relative Error Bound: 𝒔 ,𝒄 ,𝒕 Relative Error Bound: 𝒔 ,𝒄 ,𝒕 Order o Order off Appr Appro ox x. . Relative Erro Relative Error r Bo Bound: und: 𝒔 𝒔 ,𝒄 ,𝒕 Relative Error Bound: 𝒔 ,𝒄 ,𝒕 𝒏 ,𝒄 𝒏 ,𝒕 𝒏 Relative Error Bound: 𝒔 𝒏 ,𝒄 𝒏 ,𝒕 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 𝒏 1 1.19 × 10 7.51 × 10 1 1 1.19 1.19× ×1 1 0 0 7.51 7.51× ×1 1 0 0 2 3.14 × 10 5.54 × 10 2 2 3.14 3.14× ×1 1 0 0 5.54 5.54× ×1 1 0 0 3 2.13 × 10 4.89 × 10 3 3 2.13 2.13× ×1 1 0 0 4.89 × 10 4.89 × 10 4 1.78 × 10 4.72 × 10 4 1.78 × 10 4.72 × 10 4 1.78 × 10 4.72 × 10 5 1.66 × 10 4.80 × 10 5 1.66 × 10 4.80 × 10 5 1.66 × 10 4.80 × 10 6 1.64 × 10 5.05 × 10 6 1.64 × 10 5.05 × 10 6 1.64 × 10 5.05 × 10 8 1.79 × 10 5.99 × 10 8 8 1.79 1.79× ×1 1 0 0 5.99 5.99× ×1 1 0 0 10 2.14 × 10 7.54 × 10 10 10 2.14 2.14× ×1 1 0 0 7.54 7.54× ×1 1 0 0 12 2.69 × 10 9.85 × 10 12 12 2.69 2.69× ×1 1 0 0 9.85 9.85× ×1 1 0 0 16 4.71 × 10 1.81 × 10 16 16 4.71 4.71× ×1 1 0 0 1.81 1.81× ×1 1 0 0 3.2.4. Notes 3. 3.2. 2.4. No 4. Notes tes AppliedMath 2023, 3 356 3.2.4. Notes The form of the approximation, as stated in Theorem 2, for arcsine: 2n s (y) = 1 y d y (51) n å n,k k=0 is consistent with the optimum Padè approximant form speciﬁed by Abramowitz [14] and stated in Equation (15). The relative error bound for the Abramowitz approximation is 3.04 10 . The relative error bound for the 4th order approximation, s , as speciﬁed by Equation (A27), is 1.78 10 whilst a ﬁfth order approximation, s , has a relative error bound of 1.66 10 . A comparison of the results detailed in Tables 1 and 2 indicate that the approxima- tions, as stated in Corollary 1, are more accurate than those speciﬁed in Theorem 2. For comparison, the fourth order approximations, s , for arcsine have the respective relative 6 5 error bounds of 2.49 10 and 1.78 10 . 3.3. Alternative Approximations II: Integration of Arcsine The integral of arcsine, e.g., [8] (4.26.14, p. 122), is: asin(l)dl = yasin(y) + 1 y 1, jyj 1 (52) which implies Z q asin(y) = asin(l)dl + 1 1 y (53) There is potential with this relationship, and based on approximations to arcsine that are integrable, to deﬁne new approximations to arcsine, with a lower relative error bound, than the approximations detailed in Corollary 1 and Theorem 2. The approximations to arcsine, as deﬁned by s , in Theorem 2, are integrable and lead to the following approximations. Theorem 3. Alternative Approximations II—Integration of Arcsine. Alternative approximations, of order n, n 2 0, 1, 2, , for arcsine, arccosine and arctan- f g gent, are: 2 3 1+ 6 7 p 2n 2 1 1 y 6 7 asin(y) s (y) = 61 1 y + å d 7 n,k y4 2 + k 5 k=0 (54) " # 2+k 2n p 1 1 y asin(y) s (y) = p 1 y + d n,k 2 2 + k 1 y k=0 2 3 1+ 6 7 p 2n 2 1 1 y p 1 6 7 acos(y) c (y) = 61 1 y + å d 7 n,k 2 y 2 + k 4 5 k=0 (55) " # 2+k 2n 1 1 y acos y c y = p 1 y + d ( ) ( ) å n n,k 2 + k 1 y k=0 AppliedMath 2023, 3 357 2 2 33 2 2n 6 6 77 1 + y 1 1 n,k 6 6 77 atan(y) t (y) = 1 p + 1 4 4 55 2 k y 2 + k 1 + y k=0 1+ (56) (1 + y ) " " ## 2+k 2n p y y n,k A 2 atan(y) t (y) = 1 + y 1 p + 1 1+k/2 2 2 + k 2 1 + y k=0 [1 + y ] (k + 1)c n,k+1 where d = with c being deﬁned by Equation (29). n,k n,k Proof. Consider the approximation for arcsine deﬁned by s and stated in Theorem 2. Use of this approximation in Equation (53) leads to q Z 2n k/2 asin(y) [1 1 y + d t 1 t dt (57) å n,k k=0 The result 1+k/2 k/2 1 1 y t 1 t dt = (58) 2 + k leads to the approximation s deﬁned in Equation (54). The alternative approximations " # h i p y follow according to c (y) = s (y), c (y) = s 1 y , t (y) = s p , n n n n n 1 + y h i p p A A 2 2 s (y) = c 1 y and t (y) = c [1/ 1 + y . n n n n 3.3.1. Explicit Approximations for Arcsine A second order approximations for arcsine is: h i 2 2 16 16p 3p 1 32 8p 15p s (y) = + 1 1 y + + y + 3 5 2 y 3 5 4 2 2 26 45p 20 5p 5p 3 5 (59) + 6p y + + y + 3 16 9 4 8 p p 32p 14 16p 3p 2 3 2 2 9 + 3p y 1 y + + y 1 y 5 3 5 2 and has a relative error bound of 1.56 10 . A fourth order approximation has a relative error bound of 1.00 10 . 3.3.2. Results The relative error bounds associated with the approximations s (y), c (y) and t (y) n n to arcsine, arccosine and arctangent, as speciﬁed by Theorem 3, are detailed in Table 3. The A A relative errors associated with s y , c y and t y become unbounded, respectively, at ( ) ( ) ( ) n n the points zero, one and zero. The graphs of the relative errors for s (y) and s (y) are shown in Figure 10. AppliedMath 2022, 2, FOR PEER REVIEW 14 Proof. Consider the approximation for arcsine defined by 𝑠 and stated in Theorem 2. Use of this approximation in Equation (53) leads to asin (𝑦) ≈ [1 − 1−𝑦 + 𝑑 [𝑡(1 − 𝑡 ) ] 𝑑𝑡 (57) The result 1−(1 −𝑦 ) (58) 𝑡(1 − 𝑡 ) 𝑑𝑡 = 2+𝑘 leads to the approximation 𝑠 defined in Equation (54). The alternative approximations follow according to 𝑐 (𝑦 ) = −𝑠 (𝑦 ), 𝑐 (𝑦 ) =𝑠 1−𝑦 , 𝑡 (𝑦 ) =𝑠 , 𝑠 (𝑦 ) = 𝑐 1−𝑦 and 𝑡 (𝑦) = 𝑐 [1/ 1+ 𝑦 . □ 3.3.1. Explicit Approximations for Arcsine A second order approximations for arcsine is: AppliedMath 2023, 3 358 16 16𝜋 3𝜋 1 −32 8𝜋 15𝜋 𝑠 (𝑦) = + − ⋅ ⋅1− 1− 𝑦 + − + 𝑦 + 3 5 2 𝑦 3 5 4 26 45𝜋 −20 5𝜋 5𝜋 +6𝜋 − 𝑦 + − + 𝑦 + (59) Table 3. Relative error bounds associated with the approximations, speciﬁed in Theorem 3, for 3 16 9 4 8 arcsine, arccosine (interval [0, 1]) and arctangent (interval [0, ¥)). 32𝜋 −14 16𝜋 3𝜋 9 + −3𝜋 𝑦 1− 𝑦 + − + 𝑦 1− 𝑦 5 3 5 2 Order of Approx. Relative Error Bound: s (y), c (y), t (y) n n 0 0.145 and has a relative error bound of 1.56 × 10 . A fourth order approximation has a relative 1 2.63 10 error bound of 1.00 × 10 . 2 1.56 10 3 1.18 10 3.3.2. Results 4 1.00 10 5 9.22 10 The relative error bounds associated with the approximations 𝑠 (𝑦) , 𝑐 (𝑦) and 8.91 10 𝑡 (𝑦) to arcsine, arccosine and arctangent, as specified by Theorem 3, are detailed in Table 3. 8 9.19 10 The relative errors associated with 𝑠 (𝑦) , 𝑐 (𝑦) and 𝑡 (𝑦) become unbounded, respec- 10 1.03 10 12 1.23 10 tively, at the points zero, one and zero. The graphs of the relative errors for 𝑠 (𝑦) and 16 1.95 10 𝑠 (𝑦) are shown in Figure 10. Figure 10. Graph of the relative errors in the approximations, as deﬁned in Theorem 3, to arc-sin. Figure 10. Graph of the relative errors in the approximations, as defined in Theorem 3, to arc-sin. 3.4. Alternative Approximations Table 3. Relative error bounds associated with the approximations, specified in Theorem 3, for Alternative approximations can be determined. For example, the relationship: arcsine, arccosine (interval [0,1]) and arctangent (interval [0,∞)). Z q 2 2 asin(l) dl = 2y + 2 1 y asin(y) + yasin(y) (60) Order of Approx. Relative Error Bound: 𝒔 (𝒚) , 𝒄 (𝒚) , 𝒕 (𝒚) 𝒏 𝒏 𝒏 0 0.145 leads to a quadratic equation for arcsine when an integrable approximation for asin(y) is utilized. As a second example, the relationship Z q y asin(y) 2 2 1 l dl = 1 y + (61) 2 2 implies Z q 2 2 asin(y) = 2 1 l dl y 1 y (62) and, thus, an approximation for arcsine can be determined when a suitable approximation for 1 y , which is integrable, is available. 4. Error and Convergence Consider the definition of the square of the radial function r as defined by Equation (18) and the error # in the nth order approximation, f , to r , as deﬁned in Theorem 1, i.e., n n 2 2 r (y) = acos(y) + y = f (y) + # (y), 0 y 1. (63) n n AppliedMath 2023, 3 359 Consistent with the nature of a nth order two point spline approximation based on the (k) (k) points zero and one, it is the case that # (0) = # (1) = 0, k 2 f0, 1, . . . , ng. n n From Equation (63) it follows that q q 2 2 acos(y)= f (y) y + # (y) = f (y) y + d (y) = c (y) + d (y) n n n c,n n c,n q q (64) 2 2 d (y)= f (y) y + # (y) f (y) y c,n n n n where c (y) = f (y) y is the nth order approximation to arccosine deﬁned in Corollary n n 1 and the error in this approximation is d y . For y ﬁxed, and for the convergent case ( ) c,n where lim # (y) = 0, it is the case that lim d (y) = 0. Hence, for y ﬁxed, convergence of n c,n n!¥ n!¥ n!¥ f y to r y as n increases, is sufﬁcient to guarantee the convergence of c y to acos y . ( ) ( ) ( ) ( ) n n Consider the nth order approximation to arcsine, s (y) = p/2 c (y), as given in n n Corollary 1. It then follows that asin(y) = c (y) d (y) = s (y) d (y) (65) n c,n n c,n Again, for y fixed, a sufficient condition for convergence of s (y) to asin(y) is for lim # (y) = 0. n n n!¥ h i As atan(y) = acos 1/ 1 + y , it follows that the error d (y) in the approximation, t,n t , to arctangent, as given by Corollary 1, yields the relationship " # " # 1 1 p p atan(y) = t (y) + d (y) = c + d (66) n t,n n c,n 2 2 1 + y 1 + y h i and, thus, d (y) = d 1/ 1 + y . Again, for y ﬁxed, convergence of t (y) to atan(y) is t,n c,n n guaranteed if lim # (y) = 0. n!¥ The goal, thus, is to establish convergence of the approximations specified by Theorem 1, i.e., to show that lim # (y) = 0. To achieve this goal, the approach is to determine a series n!¥ for the error function # and this can be achieved by ﬁrst establishing a differential equation for # . 4.1. Differential Equation for Error Consider Equation (64): acos(y) = f (y) + # (y) y , y 2 [0, 1]. Differentia- n n tion yields (1) (1) 1 f (y) + # (y) 2y n n p = p , y 2 [0, 1) (67) 2 2 1 y 2 f (y) + # (y) y n n and after squaring and simpliﬁcation the equation becomes h i h i (1) (1) 2 2 4 f (y) + # (y) y = 1 y f (y) + # (y) 2y (68) n n n n Rearrangement leads to the differential equation for the error function: h i h i (1) (1) (1) 2 2 1 y # (y) + 2 1 y f (y) 2y # (y) 4# (y)+ n n n n (69) h i (1) 2 2 1 y f (y) 2y 4 f (y) y = 0, # (0) = 0. n n n A polynomial expansion can be used to solve for # (y). Theorem 4. Polynomial Form for Error Function. AppliedMath 2023, 3 360 A polynomial form for the error function, # , as deﬁned by the differential equation speciﬁed in Equation (69), is 2n+1 ¥ k k e (y) = [C C ]y + C y , n 2 f3, 4, . . .g (70) n å k,k n,k å k,k k=n+1 k=2n+2 (k) where C is the kth coefﬁcient deﬁned in Theorem 1 and C = f (0)/k!. n,k k,k Proof. The proof is detailed in Appendix E. Explicit Approximations Polynomial expansions for e , of orders three and four, are: 964 62p 35p 994 1837p 4 2 5 e (y) = + y + + 21p y + 45 3 4 15 40 2584 110p 35p 785 3401p 6 2 7 (71) + y + + 5p y + 45 3 2 45 336 8 9 10 11 4y 35py 128y 63py + + . . . 35 1152 1575 2816 8704 2903p 63p 98, 176 692p 5 2 6 e (y) = + y + + 105p y + 105 40 2 315 3 45, 056 32, 205p 5504 315p 2 7 8 (72) + 135p y + + 164p y + 105 112 21 4 2 10 11 18, 944 41, 315p 35p 128y 63py + y + + . . . 315 1152 2 1575 2816 4.2. Convergence (n) First, consistent with Equation (22), C = f (0)/n!. Second, consistent with n,n Equation (26), it is the case that (1 2/n) C = C (73) n,n n,n2 1 1/n ( ) As discussed in Section 3.1.1, it is the case that jC j < jC j and with jC j < 1 n,n n,n2 n,n for n > 2. It then follows that lim C = 0 and the decrease in magnitude is mono- n,n n!¥ tonic as n increases for even and odd values. Third, from Equation (70) and the result jC j < jC j = jC j, it follows, for the case of 0 < y < 1, y ﬁxed, that n,n n,n2 n2,n2 2n+1 ¥ k k je (y)j C C y +jC j y å å n k,k n,k 2n+2,2n+2 k=n+1 k=2n+2 " # (74) 2n+2 2n+1 n+1 y å C C +jC j 2n+2,2n+2 k,k n,k 1 y k=n+1 2n+1 The graph of S = C C is shown in Figure 11. As this is bounded, and as n å k,k n,k k=n+1 0 < y < 1, it follows that lim e y = 0 for 0 < y < 1. ( ) n!¥ AppliedMath 2022, 2, FOR PEER REVIEW 17 −8704 2903𝜋 63𝜋 98176 692𝜋 𝑒 (𝑦) = − + 𝑦 + + − 105𝜋 𝑦 + 105 40 2 315 3 −45056 32205𝜋 5504 315𝜋 − + 135𝜋 𝑦 + + 164𝜋 − 𝑦 + (72) 105 112 21 4 −18944 41315𝜋 35𝜋 128𝑦 63𝜋𝑦 − + 𝑦 + − + ⋯ 315 1152 2 1575 2816 4.2. Convergence () First, consistent with Equation (22), 𝐶 =𝑓 (0)/𝑛! . Second, consistent with Equa- tion (26), it is the case that (1 − 2/𝑛) 𝐶 = ⋅𝐶 (73) , , (1 − 1/𝑛) As discussed in Section 3.1.1, it is the case that 𝐶 <𝐶 and with 𝐶 <1 for , , , 𝑛> 2 . It then follows that 𝑙𝑖𝑚 𝐶 =0 and the decrease in magnitude is monotonic as 𝑛 increases for even and odd values. Third, from Equation (70) and the result 𝐶 < 𝐶 = 𝐶 , it follows, for the case of 0<𝑦 < 1 , 𝑦 fixed, that , , | | 𝑒 (𝑦) ≤ 𝐶 −𝐶 𝑦 +𝐶 𝑦 , , , (74) ≤𝑦 𝐶 −𝐶 + 𝐶 ⋅ , , , 1−𝑦 The graph of 𝑆 = 𝐶 −𝐶 is shown in Figure 11. As this is bounded, and as , , AppliedMath 2023, 3 361 ( ) 0<𝑦 < 1 , it follows that 𝑙𝑖𝑚 𝑒 𝑦 =0 for 0<𝑦 < 1 . Figure 11. Graph of S for the case of n 2 f1, 2, . . . , 50g. Figure 11. Graph of S n for the case of 𝑛∈ {1,2, ...,50}. 5. Direct Approximation for Arctangent 5. Direct Approximation for Arctangent The approximations for arctangent detailed in Corollary 1, Theorem 2 and Theorem 3 The approximations for arctangent detailed in Corollary 1, Theorem 2 and Theorem are indirectly established. Direct approximations for arctangent can be established by 3 are indirectly established. Direct approximations for arctangent can be established by utilizing the fundamental relationships atan(y) + acot(y) = /2, acot(y) = atan(1/y) utilizing the fundamental relationships atan (𝑦) + acot (𝑦) = π/2 , acot (𝑦) = atan (1/𝑦) which implies which implies atan y = atan 1/y , y > 0 (75) ( ) ( ) 2π atan(𝑦 ) = −atan(1/𝑦 ), 𝑦 > 0. (75) 5.1. Approximations for Arctangent The following theorem details a spline based approximation for arctangent. 5.1. Approximations for Arctangent The following theorem details a spline based approximation for arctangent. Theorem 5. Approximations for Arctangent. Given a nth order spline based approximation, g (y), for atan(y), 0 y 1, based on the Theorem 5. Approximations for Arctangent. points zero and one, it is the case that Given a 𝑛𝑡ℎ order spline based approximation, g (𝑦 ), for 𝑎𝑡𝑎𝑛 (𝑦 ), 0≤𝑦 ≤ 1 , based on the points zero and one, it is the case tha< t g (y), 0 y 1 atan(y) (76) : g , y > 1 2 y The resulting nth order approximation, t , n 2 f0, 1, 2, . . .g, for arctangent is 2 2n+1 > d y + d y + . . . + d y , 0 y 1 n,1 n,2 n,2n+1 t (y) = (77) d d d n,1 n,2 n,2n+1 : . . . , 1 < y < ¥ 2 2n+1 2 y y where the coefﬁcients d , i 2 1, . . . , 2n + 1 , are deﬁned according to: f g n,i ir (1) (n + 1)! > a , 1 i n å n,r (r + n + 1 i)!(i r)! > r=0 < ir (1) (n + 1)! d = a + (78) n,i å n,r (r + n + 1 i)!(i r)! > r=in1 in1 > n > (1) r! : b , n + 1 i 2n + 1 n,r (r + n + 1 i)!(i n 1)! r=in1 Here: ru r (ru) r (ru) g (0) (n + u)! (1) g (1) (n + u)! a = , b = (79) n,r n,r å å (r u)! u!n! (r u)! u!n! u=0 u=0 where g y = atan y and ( ) ( ) 0, k 2 f0, 2, 4, . . .g (k) g (0) = (80) (k1)/2 (1) (k 1)!, k 2 f1, 3, 5, . . .g AppliedMath 2023, 3 362 , k = 0 > 4 k1 >(1) , k = 1, 2, 3 0, k 2 f4, 8, . . .g (k) g (1) = (81) k 1 k 2 ( )( ) (k2) g (1), k 2 f5, 9, 13, . . .g > 2 (k1) (k 1)g (1), k 2 f6, 10, 14, . . .g (k 1) (k1) g (1), k 2 f7, 11, 15, . . .g Proof. Consider the approximation g y for atan y , 0 y 1. The relationship ( ) ( ) atan(y) = p/2 atan(1/y) implies p 1 p 1 atan(z) = atan g (y), z > 1, y = , 0 y < 1. (82) 2 z 2 z (k) (k) The formulas for g (0), g (1) and d can be established in a manner consistent with the n,i nature of the proof detailed in Appendix F. 5.1.1. Analytical Approximations Approximations for arctangent, of orders zero to two, are: py , 0 y 1 t (y) = (83) p p , y > 1 2 4y 5 3p 3 p > 2 3 y y + y , 0 y 1 2 4 2 2 t (y) = (84) >p 1 5 3p 1 3 p 1 : + , y > 1 2 3 2 y 2 4 y 2 2 y 33 5p 15p 19 3p 3 4 5 y y + 12 y y , 0 y 1 4 2 4 4 2 t (y) = (85) p 1 33 5p 1 15p 1 19 3p 1 + 12 + , y > 1 3 4 5 2 y 4 2 y 4 y 4 2 y Approximations, of orders three and four, are detailed in Appendix G. 5.1.2. Approximations for Arccosine and Arcsine " # h i The relationships asin(y) = atan p , y 2 [0, 1) and acos(y) = atan 1 y /y , 1 y y 2 (0 , 1], imply the following approximations for arcsine and arccosine: 2 2n+1 d y d y d y 1 n,1 n,2 n,2n+1 p + + . . . + , 0 y p > 2 n+1/2 < 1 y 2 1 y 2 1 y ( ) s (y) = (86) n+1/2 2 2 > 2 > p d 1 y d 1 y d 1 y 1 n,2 n,1 n,2n+1 . . . , p < y 1 2 2n+1 2 y y 2 AppliedMath 2023, 3 363 2 2n+1 p d y d y d y 1 n,1 n,2 n,2n+1 p p > . . . , 0 y 2 n+1/2 < 2 2 (1 y ) 1 y 2 (1 y ) c (y) = (87) n+1/2 2 2 > 2 d 1 y d 1 y > d 1 y 1 n,2 n,2n+1 n,1 : p + + . . . + , y < 1 2 2n+1 y y y Alternative approximations for arcsine and arccosine speciﬁed according to A A s (y) = p/2 acos(y) and c (y) = p/2 asin(y) lead to identical expressions, i.e., n n A A s (y) = s (y) and c (y) = c (y). n n n n As an example, the third order approximation for arcsine is 55 35p 265 4 5 y 21p y y y > 2 4 4 > p + 3/2 5/2 > 2 2 2 2 > 1 y 1 y 3[1 y ] [ ] [1 y ] 331 35p 63 6 7 > y 5p y 6 2 4 1 + , 0 y p > 7/2 2 2 1 y 2 [ ] [1 y ] s (y) = (88) > 55 35p 265 2 5/2 > 2 2 3/2 1 y 21p 1 y > 2 1 y > p 1 y 2 4 4 > + + 3 5 2 y 3y y y 331 35p 63 2 2 1 y 5p 1 y 6 2 4 1 : + , p < y < 1 6 7 y y 5.1.3. Results The relative errors associated with the approximations for arctangent, of orders one to six, are shown in Figure 12. The relative error bounds associated with the approximations to arctangent, arcsine and arccosine are detailed in Table 4. The relative error bound associate with the third order approximation for arcsine, as speciﬁed by Equation (88), is 3.73 10 AppliedMath 2022, 2, FOR PEER REVIEW 20 which is comparable with the third order approximation speciﬁed in Corollary 1 whose relative error is 2.84 10 . Figure 12. Graphs of the relative errors in approximations, of orders 1 to 6, for arctangent as deﬁned Figure 12. Graphs of the relative errors in approximations, of orders 1 to 6, for arctangent as defined in Theorem 5. in Theorem 5. Table 4. Relative error bounds, associated with the approximations detailed in Theorem 5 and The- orem 6 for arcsine, arccosine and arctangent. The interval [0,1] is assumed for arcsine and arccosine; the interval [0,∞] for arctangent. Theorem 6—Relative Error Bound for Order of Theorem 5—Relative Error Arctangent. The Value Assumed for Spline Bounds: 𝒔 ,𝒄 ,𝒕 𝜹 is the Second Value Stated in 𝒏 𝒏 𝒏 𝒏,𝟎 Approx. Equation (92). 0 2.15 × 10 1 2.18 × 10 4.31 × 10 2 1.68 × 10 3.21 × 10 3 3.73 × 10 6.77 × 10 4 3.34 × 10 6.34 × 10 5 6.39 × 10 1.17 × 10 6 6.22 × 10 1.10 × 10 8 1.82 × 10 3.09 × 10 10 3.74 × 10 6.06 × 10 5.2. Improved Approximation: Use of Integral for Arctangent Consider the known integral 𝑦 atan(𝑦) atan(𝑦) 𝑦 atan(𝜆 𝜆 )𝑑𝜆 = + − (89) 2 2 2 which implies 2 𝑦 𝑎𝑡𝑎𝑛( 𝑦) = ⋅ + a 𝜆 tan (𝜆 )𝑑𝜆 . (90) 1+ 𝑦 2 An integrable approximation for 𝑦atan(𝑦) , for [0,1], leads to an approximation for arctan- gent. Theorem 6. Improved Approximations for Arctangent. The 𝑛 th order approximation for arctangent, based on Equation (90), is defined according to δ 𝑦 δ 𝑦 δ 𝑦 , , , ⎡ ⎧ ⎤ + +⋯ + , 0 ≤ 𝑦 ≤ 1 3 4 2𝑛 + 3 ⎢ ⎪ ⎥ 2 𝑦 𝜋(𝑦 −1) 1 ⎢ ⎥ 𝑡 (𝑦 ) = + δ + −δ (𝑦− 1 ) −δ ln(𝑦 ) −δ 1 − − (91) , , , , ⎢ ⎥ 1+ 𝑦 2 4 𝑦 ⎢ ⎥ ⎪ δ 1 δ 1 , , ⎢ ⎪ ⎥ 1 − −⋯ − 1 − , 1 < 𝑦 < ∞ ⎣ ⎩ 2 𝑦 2𝑛 − 1 𝑦 ⎦ AppliedMath 2023, 3 364 Table 4. Relative error bounds, associated with the approximations detailed in Theorem 5 and Theorem 6 for arcsine, arccosine and arctangent. The interval [0, 1] is assumed for arcsine and arccosine; the interval [0,¥] for arctangent. Order of Theorem 6—Relative Error Bound for Arctangent. Theorem 5—Relative Spline The Value Assumed for d is the Second Value n,0 Error Bounds : s ,c ,t n n n Approx. Stated in Equation (92). 0 2.15 10 2 3 2.18 10 4.31 10 3 4 2 1.68 10 3.21 10 5 6 3 3.73 10 6.77 10 5 6 3.34 10 6.34 10 6 6 5 6.39 10 1.17 10 7 7 6 6.22 10 1.10 10 8 9 1.82 10 3.09 10 10 11 10 3.74 10 6.06 10 5.2. Improved Approximation: Use of Integral for Arctangent Consider the known integral y atan(y) atan(y) y latan(l)dl = + (89) 2 2 2 which implies 2 y atan(y) = + latan(l)dl . (90) 1 + y 2 An integrable approximation for yatan(y), for [0, 1], leads to an approximation for arctangent. Theorem 6. Improved Approximations for Arctangent. The nth order approximation for arctangent, based on Equation (90), is deﬁned according to 2 3 3 4 2n+3 d y d y d y > n,1 n,2 n,2n+1 + + . . . + , 0 y 1 6 > 7 3 4 2n + 3 6 > 7 6 > 7 < 2 6 7 2 y p y 1 1 6 7 t (y) = + d + d (y 1) d ln(y) d 1 (91) n,0 n,1 n,2 n,3 26 7 1 + y 2 > 4 y 6 > 7 6 > 7 4 > 5 d 1 d 1 n,4 n,2n+1 1 . . . 1 , 1 < y < ¥ 2 2n1 2 y 2n 1 y where the coefﬁcients d are deﬁned in Equation (78) and n,i p 1 d d d n,1 n,2 n,2n+1 d = or d = + + . . . + (92) n,0 n,0 4 2 3 4 2n + 3 Here d is associated with atan()d and with the ﬁrst value being exact. The second value n,0 yields a lower relative error bound for the interval (1, ¥). Proof. The approximations for arctangent, as deﬁned in Theorem 5, when used in the integral in Equation (90), lead to the approximations speciﬁed by Equation (91). AppliedMath 2023, 3 365 5.2.1. Explicit Expressions Explicit approximations for arctangent, of orders one and two, are: 1 2y 5 3p 3 p 4 5 y + y + y , 0 y 1 < 2 3 4 8 5 5 1 + y t (y) = (93) > 1 59 27p 3 p 3p py + + y + 5 ln(y) + , y > 1 60 40 y 2 2 1 + y 1 2y 33 5p 19 3p 5 6 7 y + p y + 4 y y , 0 y 1 > 2 1 + y 3 10 4 14 7 2 3 1016 18p 19 1 15p 1 t (y) = (94) p + 12 6 3 2 7 > 1 105 7 6 y 4 y 6 7 , y > 1 2 4 5 1 + y 33 1 py 5p y + 2 y 2 Third and fourth order approximations are detailed in Appendix H. Explicit approximations for arcsine and arccosine can be deﬁned by utilizing the h i h i p p 2 2 relationships asin(y) = atan y/ 1 y and acos(y) = atan 1 y /y . 5.2.2. Results The relative error bounds associated with the approximations to arctangent are detailed in Table 4 and the improvement over the original approximations is evident. 6. Improved Approximations via Iteration Given an initial approximating function h for the inverse, f , of a function f , the ith iteration of the classical Newton-Raphson method of approximation leads to the ith order approximation f [h (y)] y i1 h y = h y , h y known, i 2 1, 2, . . . . (95) ( ) ( ) ( ) f g i i1 0 (1) f [h (y)] i1 6.1. Newton-Raphson Iteration: Approximations and Results for Arcsine The arcsine case is considered: An initial approximation to arcsine of h y = s y n 2 0, 1, 2, . . . , as specified by Corollary 1, Theorem 2, Theorem 3 or Section 5.1.2, ( ) ( ) f g 0 n leads to the ith order iterative Newton-Raphson approximation: sin[h (y)] y i1 h (y) = h (y) , h (y) = s (y), 0 n i i1 cos[h (y)] i1 (96) = h (y) tan[h (y)] + ysec[h (y)]. i1 i1 i1 Iteration of orders one and two lead to the approximations: sin s y y [ ( )] h (y) = s (y) = s (y) tan[s (y)] + ysec[s (y)] (97) 1 n n n n cos[s (y)] sin[s (y)] y sin s (y) y sin[s (y)] y cos[s (y)] n n h (y) = s (y) 2 n cos s y sin[s (y)] y [ ( )] n n cos s (y) cos[s (y)] (98) s (y) tan[s (y)]+ n n = s (y) tan[s (y)] + ysec[s (y)] tan + n n n ysec[s (y)] ysec[s (y) tan[s (y)] + ysec[s (y)]] n n n AppliedMath 2023, 3 366 The approximation arising from a third order iteration is detailed in Appendix I. Example and Results As an example, consider the second order approximation for arcsine arising from Theorem 2 and deﬁned by Equation (49): h i p p 2 3 4 h (y) = s (y) = 1 y y + d y + d y + d y , y 2 [0, 1]. (99) 0 2 2,2 2,3 2,4 2 2 The relative error bound associated with this approximation is 3.14 10 . The ﬁrst order iteration of the Newton-Raphson method yields the approximation h i p p 1 2 3 4 f (y) h (y) = 1 y y + d y + d y + d y 1 2,2 2,3 2,4 2 2 h h ii 2 2 3 4 (100) cos 1 y y + d y + d y + d y y 2,2 2,3 2,4 h h ii 2 2 3 4 sin 1 y y + d y + d y + d y 2,2 2,3 2,4 The relative error bound for this approximation, and associated with the interval [0, 1], is 2.13 10 . Second order iteration yields the approximation detailed in Equation (A62). The relative error bound associated with this approximation, for the interval [0, 1], is 5.68 10 The use of h (y) = s (y), as speciﬁed by Equation (A27), rather than 0 4 h (y) = s (y), leads to a relative error bound of 3.05 10 . 0 2 Consider the fourth order approximation, s , deﬁned by Equation (A27). A ﬁrst order iteration of the Newton-Raphson method yields the approximation 2 3 2 3 p py 2y 4 5 6 y + + d y + d y + d y + 6 4,4 4,5 4,6 7 f y g y = 1 y 2 4 3 ( ) ( ) 4 5 7 8 d y + d y 4,7 4,8 2 2 33 2 3 p py 2y 4 5 6 y + + d y + d y + d y + 4,4 4,5 4,6 4 4 55 (101) cos 1 y y 2 4 3 7 8 d y + d y 4,7 4,8 2 2 33 2 3 p py 2y 4 5 6 y + + d y + d y + d y + 4,4 4,5 4,6 6 6 77 2 4 3 sin 1 y 4 4 55 7 8 d y + d y 4,7 4,8 The relative error bound associated with this approximation is 1.44 10 . The improvement that is possible with Newton-Raphson iteration is illustrated in Table 5 where the original approximations to arcsine and arctangent, based on s (y), s (y), t (y) and t (y) as deﬁned in Theorem 2 and speciﬁed by Equations (49) and (50), are used. The quadratic convergence, with iteration, is evident. It is usual for the relative error improvement, with iteration, to be dependent on the relative error in the initial approximation. However, as the results in Table 5 indicate, the approximations of s (y) and t (y), with higher relative error bounds, lead to lower relative bounds with iteration than s (y) and t (y). This is due to the nature of the approximations. 2 AppliedMath 2023, 3 367 Table 5. Relative error bounds for Newton-Raphson iterative approximations to arcsine and arc- A A tangent and based on s (y), s (y), t (y) and t (y) as deﬁned in Theorem 2 and speciﬁed by 2 2 2 2 Equations (49) and (50). Relative Error Relative Error Relative Error Relative Error Order of Bound: Bound: Bound: Bound: Iteration A A h (y)=s (y) h (y)=s (y) h (y)=t (y) h (y)=t (y) 0 2 0 0 2 0 2 2 3 4 4 3 0 3.14 10 5.54 10 5.54 10 3.14 10 7 7 6 7 2.13 10 6.52 10 1.31 10 4.26 10 15 12 11 14 2 5.68 10 1.43 10 1.15 10 4.55 10 29 24 21 27 3 1.31 10 7.98 10 1.03 10 1.68 10 59 46 42 54 4 7.27 10 2.68 10 9.00 10 2.39 10 117 91 82 108 5 2.29 10 3.13 10 7.10 10 4.95 10 7. Applications 7.1. Approximations for a Set Relative Error Bounds: Arcsine With the requirement of a set relative error bound in an approximation for arsine, arccosine or arctangent, an approximation form and a set order of approximation can be speciﬁed. The following details examples of approximations for arcsine and the interval [0, 1] is assumed. For a relative error bound close to 10 , the approximation u 3/2 2 2 2 2 u p 1 y + 1 y + c 1 y + c 1 y + 2,3 2,4 s (y) = 4 (102) 5/2 c 1 y 2,5 as given by Corollary 1, yields a relative error bound of 1.81 10 . The approximation, s , deﬁned by Equation (59) yields a relative error bound of 1.56 10 . For a relative bound close to 10 , the approximation s y = c y , where c is ( ) ( ) 4 4 4 deﬁned by Equation (A22), is 2 3 4 p p py y 2 5 6 7 8 9 s y = py + y + + c y + c y + c y + c y + c y (103) ( ) 4 4,5 4,6 4,7 4,8 4,9 2 4 6 3 and has a relative error bound of 2.49 10 . The approximation deﬁned by h i s (y) = c 1 y (see Equation (A14)) is 4,1 4,1 2 3 2 s 3 p p u 6 7 4 5 u 10 1 + 2 + 2 + 2 1 + 1 y u 6 7 q 10 u 6 7 t 6 7 s (y) = 4 2 2 + 2 + 2 1 + 1 y 1 + (104) 6 7 4,1 2 s 3 3/2 6 7 6 p p 7 4 2 5 4 5 3 2 + 2 + 2 + 2 1 + 1 y and has a relative error bound of 1.19 10 . The approximation given by Abramowitz, as stated in Equation (15), has a relative error bound of 3.04 10 . If a high accuracy approximation is required then two approaches can be used. First, higher order approximations as speciﬁed in Corollary 1, Theorem 2, Theorem 3 and Theorem 5 can be used. For example, the ﬁfteenth order approximation, s , for arcsine detailed in Corollary 1 yields a relative error bound of 4.74 10 . Second, iterative approaches can be used. For example, the second order approximation, s , for arcsine arising from Theorem 2 and deﬁned by Equation (49) and a second order iteration leading to Equation (A62) has a relative error bound of 5.68 10 . An alternative approxi- mation can be deﬁned by utilizing the zero order spline approximation, as speciﬁed by AppliedMath 2023, 3 368 Equation (117), and the sixth and seventh order approximations (the function f ) which 0,6,7 yields a relative error bound of 7.65 10 (see Table 6). 7.2. Upper and Lower Bounds for Arcsine, Arccosine and Arctangent Lower, L, and upper, U, bounds for arcsine, i.e., L(y) < asin(y) < U(y), 0 < y < 1, (105) lead to the following lower and upper bounds for arccosine and arctangent: p p U(y) < acos(y) < L(y), 0 < y < 1, (106) 2 2 " # " # y y L p < atan(y) < U p , 0 < y < ¥. (107) 2 2 1 + y 1 + y 7.2.1. Published Bounds for Arcsine There is interest in upper and lower bounds for arcsine, e.g., [17–21]. The classic upper and lower bounded functions for arcsine are deﬁned by the Shafer-Fink inequality [13]: 3y py p asin(y) p , 0 y 1. (108) 2 2 2 + 1 y 2 + 1 y The relative error bound associated with the lower bounded function is 4.51 10 ; the relative error bound associated with the upper bounded function is 4.72 10 . Zhu [20] (eqn. 1.8), proposed the bounds: p 2 2 p p p p 1 + y 1 y 6 1 + y 1 y p 2 2 p p asin y p , 0 y 1 (109) ( ) p p 4 + 1 + y + 1 y 2(4 p) + 1 + y + 1 y p 2 2 where the lower relative error bound is 2.27 10 and the upper relative error bound is 5.61 10 . Zhu [21] (Theorem 1), proposed the bounds n1 2n+1 2i+1 a y + a y asin(y) n å 2 + 1 y i=0 (110) n1 2n+1 2i+1 p b y + a y , n 2 f2, 3, . . .g, 0 y 1, n i 2 + 1 y i=0 i1 i1 1 (2i 1)!! 2 i! a = 3, a = , b = p a (111) 0 i i å k i1 2i + 1 2 i! i[(2i 1)!!] k=0 The lower bound is equivalent to the bound proposed by Maleševí et al. [19] (eqn. 21). The relative errors in the bounds are low for y 1 but increase as y increases. For the case of n = 4 the relative error bound for the lower bounded function is 0.0324; for the upper bounded function the relative error bound is 0.0159. 7.2.2. Proposed Bounds for Arcsine and Arccosine Consider the approximations deﬁned in Corollary 1 and whose relative errors are shown in Figure 5. As the graphs in this ﬁgure indicate, the approximations are either upper or lower bounds for arcsine and arccosine and this is conﬁrmed by numerical analysis (for the orders considered) which shows that there are no roots, in the interval (0,1), for the error AppliedMath 2023, 3 369 function associated with the approximations. The evidence is that the approximations, s , of orders 0, 2, 4, . . . , are lower bounds for arcsine whilst the approximations of orders 1, 3, 5, . . . are upper bounds. Thus, for example, second, s , and third, s , order approximations, 2 3 as deﬁned in Corollary 1, yield the inequalities p p 2 3 4 5 py + y + c y + c y + c y asin y ( ) 2,3 2,4 2,5 2 4 (112) 2 3 p p py 2 4 5 6 7 py + y + c y + c y + c y + c y 3,5 3,6 3,7 3,4 2 4 6 for y 2 [0, 1], where, as detailed in Table 1, the lower relative error bound is 3.64 10 and the upper relative error bound is 2.84 10 . It then follows, from Equation (106), that 2 3 p py 2 4 5 6 7 py + y + c y + c y + c y + c y acos(y) 3,5 3,6 3,7 3,4 4 6 (113) 2 3 4 5 py + y + c y + c y + c y 2,3 2,4 2,5 for y 2 [0, 1]. An analytical proof that the approximations for arcsine and arccosine, as detailed in Corollary 1, are upper/lower bounds is an unsolved problem. 7.2.3. Upper/Lower Bounds for Arctangent As an example of upper and lower bounds that have been proposed for arctangent, consider the bounds proposed by Qiao and Chen [22] (Theorem 3.1 and Theorem 4.2) for y > 0: 3p y p < atan(y) < 2 2 4 2 2 2 24 p + 432 24p + p 12p(12 p )y + 36p y (114) 3p y 2 2 4 2 2 2 24 p + 576 192p + 16p 12p(12 p )y + 36p y 2 4 p 64 + 735y + 945y 64 1856 + < atan(y) < 2 4 11 13 2 15y[15 + 70y + 63y ] 43, 659y 464, 373y (115) 2 4 p 64 + 735y + 945y 64 2 4 11 2 15y[15 + 70y + 63y ] 43, 659y The lower bounded function in Equation (114) has a relative error bound of 0.0520; the upper bounded function has a relative error bound of 0.0274. The error in the upper and lower bounded functions speciﬁed in Equation (115) diverges as y ! 0 but converges rapidly to zero for y 1. AppliedMath 2023, 3 370 7.2.4. Proposed Bounds for Arctangent " # As atan(y) = acos p it follows, from Equation (113), that the functions t 1 + y and t deﬁned in Corollary 1 are, respectively, upper and lower bounds for arctangent, i.e., c c p p 1 p u 3,4 3,5 p + + + + 2 2 3/2 5/2 u 4 1 + y 2 2 2 1 + y 1 + y 6[1 + y ] [ ] [1 + y ] c c t 3,6 3,7 7/2 2 2 [1 + y ] [1 + y ] (116) atan(y) p p 1 c c c 2,3 2,4 2,5 p + + + + 2 3/2 2 5/2 4 1 + y 2 2 1 + y [1 + y ] [1 + y ] [1 + y ] AppliedMath 2022, 2, FOR PEER REVIEW 26 for y 2 [0, ¥). As detailed in Table 1, the relative error bound for the lower bounded 5 4 function is 1.42 10 and 1.81 10 for the upper bounded function. 7.3. Spline Approximations Based on Upper/Lower Bounds Consider upper, 𝑓 , and lower, 𝑓 , bounded functions for arcsine as illustrated in Consider upper, f , and lower, f , bounded functions for arcsine as illustrated in Figure 13. U L Figure 13. For 𝑦 fixed at 𝑦 , a spline approximation, based on the points For y fixed at y , a spline approximation, based on the points (sin(u ), u ), u = f (y ) and o o o o L o (sin(𝑢 ),𝑢 ), 𝑢 =𝑓 (𝑦 ) and (sin(𝑣 ),𝑣 ), 𝑣 =𝑓 (𝑦 ), can readily be determined. From (sin(v ), v ), v = f (y ), can readily be determined. From such an approximation, an o o o U o such an approximation, an approximation to 𝑥 =asin(𝑦 ) can then be determined. approximation to x = asin(y ) can then be determined. o o Figure 13. Illustration of upper and lower bounded approximations to arcsine and the two basis Figure 13. Illustration of upper and lower bounded approximations to arcsine and the two basis points (sin(𝑢 ), 𝑢 ), (sin(𝑣 ),𝑣 ) for two point spline based approximations. points (sin(u ), u ), (sin(v ), v ) for two point spline based approximations. o o o o Theorem 7. Spline Approximations Based on Upper/Lower Bounds. Theorem 7. Spline Approximations Based on Upper/Lower Bounds. Consider lower, 𝑓 , and upper, 𝑓 , bounded approximations for arcsine. The zero order spline Consider lower, f , and upper, f , bounded approximations for arcsine. The zero order spline approximation for arcsine, ba L sed on th Ue approximations 𝑓 and 𝑓 , is approximation for arcsine, based on the approximations f and f , is L U ( ) [ ( )] ( ) [ ( )] [ ( ) ( )] 𝑓 𝑦 𝑓 𝑦 − 𝑓 𝑦 𝑓 𝑦 +𝑦 𝑓 𝑦 − 𝑓 𝑦 𝑓 (𝑦 ) = , 𝑦 ∈ (0,1). (117) [𝑓 (𝑦) ] −𝑠𝑖𝑛 [𝑓 (𝑦) ] f (y)sin[ f (y)] f (y)sin[ f (y)] + y[ f (y) f (y)] L U U L U L f (y) = , y 2 (0, 1). (117) The 𝑛𝑡ℎ order spline approximation for arcs sin[ f (y)] s in in e, [ based o f (y)] n the approximations 𝑓 and 𝑓 , is U L [ ( ) ] 𝑠𝑖𝑛 𝑣 −𝑦 ( ) 𝑓 𝑦 = ⋅ The nth order spline approximation for arcsine, based on the approximations f and f , is L U [ ( ) ( )] 𝑠𝑖𝑛 𝑣 −𝑠𝑖𝑛 𝑢 (𝑛 + 𝑟) ! 𝑢 ⎡ ⎤ [ ( ) ( )] 𝑟! 𝑛! 𝑠𝑖𝑛 𝑣 −𝑠𝑖𝑛 𝑢 ⎢ ⎥ [𝑦 − 𝑖𝑠𝑛 (𝑢 )] . + ⎢ ( ) ⎥ 𝑓 [𝑠𝑖𝑛( 𝑢 )] (𝑛 + 𝑢)! 1 ⎢ ⎥ . . [ ( ) ( )] (𝑟 − 𝑢) ! 𝑢! 𝑛! 𝑠𝑖𝑛 𝑣 −𝑠𝑖𝑛 𝑢 ⎣ ⎦ (118) [ ( )] 𝑦− 𝑖𝑠𝑛 𝑢 [𝑠𝑖𝑛 (𝑣 ) −𝑠𝑖𝑛 (𝑢 )] (𝑛 + 𝑟) ! 𝑣 ⎡ + ⎤ 𝑟! 𝑛! [𝑠𝑖𝑛 (𝑣 ) −𝑠𝑖𝑛 (𝑢 )] ⎢ ⎥ [𝑠𝑖𝑛 (𝑣 ) −𝑦] . ( ) ⎢ ⎥ (−1) 𝑓 [𝑠𝑖𝑛( 𝑣 )] (𝑛 + 𝑢)! 1 ⎢ ⎥ . . [ ( ) ( )] (𝑟 − 𝑢) ! 𝑢! 𝑛! 𝑠𝑖𝑛 𝑣 −𝑠𝑖𝑛 𝑢 ⎣ ⎦ ( ) ( ) ( ) ( ) for 𝑦∈ 0,1 , 𝑢 =𝑓 𝑦 , 𝑣 =𝑓 𝑦 , 𝑓 𝑦 =asin(𝑦) and 1 + (−1) [ ] 𝑑 𝑘, 𝑖 𝑦 ( ) (119) 𝑓 (𝑦 ) = , 𝑘 ∈ {1,2, … } (1− 𝑦 ) where [ ] 𝑑 1,0 =1, [ ] [ ] 𝑑 2,0 = 0, 𝑑 2,1 =1, (120) 𝑑 [3,0] = 1, 𝑑 [3,1] = 0, 𝑑 [3,2] =3, and for 𝑘> 3 : 𝑑 [𝑘− 1,1 ], 𝑖 = 0 𝑘+ 𝑖− 1 1 ( ) [ ] [ ] 𝑖+ 1 𝑑 𝑘− 1, 𝑖+ 1 +2 − 𝑑 𝑘− 1, 𝑖 − 1 , 1 ≤𝑖 ≤𝑘 − 3 𝑑 [𝑘, 𝑖 ] = (121) 2 2 𝑘+ 𝑖− 1 1 2 − 𝑑 [𝑘− 1, 𝑖 − 1 ], 𝑘 − 2 ≤ 𝑖 ≤ 𝑘 − 1 2 2 𝑠𝑖𝑛 𝑠𝑖𝑛 𝑠𝑖𝑛 AppliedMath 2023, 3 371 n+1 sin v y [ ( ) ] f (y) = n+1 [sin(v ) sin(u )] o o 2 3 (n+r)!u 6 7 r!n![sin(v ) sin(u )] o o n 6 7 6 7 y sin u å [ ( )] (ru) 6 r1 7 f [sin(u )] (n + u)! 1 r=0 4 5 (ru)! u!n! [sin(v ) sin(u )] u=0 o o n+1 (118) [y sin(u )] n+1 [sin(v ) sin(u )] o o 2 3 (n+r)!v 6 7 r!n![sin(v ) sin(u )] o o 6 7 6 i 7 [sin(v ) y] å 6 7 o ru (ru) 1) f [sin(v ) 6 7 r1 o r=0 (n + u)! 1 4 5 (ru)! u!n! [sin(v ) sin(u )] u=0 o o for y 2 (0, 1), u = f (y), v = f (y), f (y) = asin(y) and 0 L 0 U " # k+i+1 1 + (1) d[k, i] y k1 (k) f (y) = , k 2 f1, 2, . . .g (119) k + i + 1 1 i=0 2 2 (1 y ) where d[1, 0] = 1, d[2, 0] = 0, d[2, 1] = 1, (120) d[3, 0] = 1, d[3, 1] = 0, d[3, 2] = 3, and for k > 3: d[k 1, 1], i = 0 k + i 1 1 (i + 1)d[k 1, i + 1] + 2 d[k 1, i 1], 1 i k 3 d[k, i] = (121) 2 2 > k + i 1 1 2 d[k 1, i 1], k 2 i k 1 2 2 Proof. The proof is detailed in Appendix J. Results Consider the approximations the approximation s , i 2 f0, 1, 2, . . .g, for arcsine as detailed in Corollary 1 where approximations, of order 0, 2, 4, . . . , are lower bounds and the approximations, of orders 1, 3, 5, . . . , are upper bounds. For example, with p p f (y) = s (y) and f (y) = s (y), s (y) = c (y), s (y) = c (y) with c and c L U 5 5 5 5 4 4 4 4 2 2 deﬁned by Equation (A22) and Equation (A23), the zero order spline approximation, as speciﬁed by Equation (117), is s (y) sin[s (y)] s (y) sin[s (y)] + y[s (y) s (y)] 4 5 5 4 5 4 f (y) = (122) 0,4,5 sin[s (y)] sin[s (y)] 5 4 AppliedMath 2023, 3 372 The relative error bound for this approximation, over the interval 0, 1 , is 8.22 10 . [ ] Other results are detailed in Table 6 and clearly show the high accuracy of the approxima- tions. Table 6. Relative error bounds, over the interval [0, 1], for spline approximations based on upper and lower bounded approximations to arcsine and as speciﬁed in Theorem 7. Upper/Lower Bounded Functions: Spline Notation for Relative Error s Deﬁned in Corollary 1 Order Approx. Bound f (y) = s (y), f (y) = s (y) 0 f 2.43 10 L 0 U 1 0,0,1 1 f 1.45 10 1,0,1 2 f 1.31 10 2,0,1 3 f 1.44 10 3,0,1 4 f 1.77 10 4, 0,1 f y = s y , f y = s y 0 f ( ) ( ) ( ) ( ) 1.41 10 L 2 U 3 0,2,3 1 f 4.48 10 1,2,3 2 f 2.05 10 2,2,3 3 f 1.14 10 3,2,3 4 f 6.98 10 4,2, 3 f (y) = s (y), f (y) = s (y) 0 f 8.22 10 L 4 U 5 0,4,5 1 f 1.48 10 1,4,5 2 f 3.78 10 2,4,5 3 f 1.16 10 3,4,5 4 f 3.95 10 4,4,5 f (y) = s (y), f (y) = s (y) 0 f 7.56 10 L 6 U 7 0,6,7 1 f 1.27 10 1,6,7 2 f 2.97 10 2,6,7 3 f 8.30 10 3,6,7 4 f 2.57 10 4,6,7 7.4. Approximations for Arcsine Squared and Higher Powers k k k There is interest in approximations for acos(y) , asin(y) , asin(y) /y , k 2 f2, 3, . . .g, e.g., [23–26]. The standard series for asin(y) , e.g., [7] (1.645.2), is 2n+1 2k 2k+2 2 [k!] y asin(y) S (y) = (123) 2,n k + 1 2k + 1 ! ( )( ) k=0 A A The nth order approximation, s , speciﬁed in Corollary 1, leads to the approximations s n 2,n for asin(y) deﬁned according to p 3/2 2 A 2 2 2 2 s (y) = p 1 y + 1 y + c 1 y + c 1 y + . . . n,3 n,4 2,n (124) n+1/2 +c 1 y n,2n+1 for n 2. The relative errors in S and s are shown in Figure 14. The approximations 2,n 2,n deﬁned by s have better overall relative error performance; in particular, they are sharp 2,n at the point one. AppliedMath 2022, 2, FOR PEER REVIEW 28 𝑠 (𝑦 ) = −𝜋 1−𝑦 + (1− 𝑦 ) +𝑐 [1− 𝑦 ] +𝑐 [1− 𝑦 ] +⋯ , , (124) +𝑐 [1− 𝑦 ] for 𝑛≥ 2 . The relative errors in 𝑆 and 𝑠 are shown in Figure 14. The approxima- , , AppliedMath 2023, 3 373 tions defined by 𝑠 have better overall relative error performance; in particular, they are sharp at the point one. Figure 14. Graph of the relative errors in approximations to the square of arcsine as given by Figure 14. Graph of the relative errors in approximations to the square of arcsine as given by Equa- tion (123) (orders 2 to 6) and Equation (124) (orders 2 to 4). Equation (123) (orders 2 to 6) and Equation (124) (orders 2 to 4). 7.4.1. Approximations for Even Powers of Arcsine 7.4.1. Approximations for Even Powers of Arcsine Based on the approximation for the square of arcsine, as speciﬁed by Equation (124), Based on the approximation for the square of arcsine, as specified by Equation (124), the following result can be stated: the following result can be stated: Theorem 8. Approximation for Even Powers of Arcsine. Theorem 8. Approximation for Even Powers of Arcsine. Based on the nth order approximation, s , speciﬁed in Corollary 1, the even powers of arcsine Based on the nth order approximation, 𝑠 , specified in Corollary 1, the even powers of arcsine can be approximated according to can be approximated according to () m(2n+1) h i 2m k/2 2m A 2 asin(y) s (y) = b 1 y / , m 2 f1, 2, . . .g (125) n å k (125) [ ] ( ) (𝑦) ≈ 𝑠 (𝑦) = 𝛽 1− 𝑦 , 𝑚 ∈ {1,2,…} k=0 where where b = c c . . . c (126) k å n,i n,i n,i 1 2 m i +i +...+i =k 1 2 m i ,i ,...,i 2 f0,1,2,...,2n+1g 𝛽 = m 𝑐 𝑐 …𝑐 1 2 , , , (126) Here,c c . . . c are deﬁned by Equation (29). n,i n,i n,i 1 2 m , ,…, ∈ {,,,…,} Here, 𝑐 ,… ,𝑐 are defined by Equation (29). A Proof., This result , follows from expansion of s to the 2mth power, i.e., h i 2n+1 2n+1 Proof. This result follows f 2m rom expansion of 𝑠 to the 2mth power, (i+i i+ .e. ..., +i )/2 A 2 s (y) = c c c 1 y (127) n å å n,i n,i n,i 2 m 21nn ++ 21 i=1 i =1 2 m () ii++…+i /2 12 m A 2 sy ()=− c c c 1 y () (127) nn ,,inini 12 , m k/2 ii== 11 1 m and collecting terms associated with 1 y . ( ) and collecting terms associated with 1−𝑦 .□ 7.4.2. Example 7.4.2. Example For example, the nth order approximation for asin(y) is For example, the 𝑛 th order approximation for 𝑎𝑠𝑖𝑛 (𝑦 ) is h i 4 1/2 A [𝑠 (𝑦) ] =𝑐 2 +2𝑐 𝑐 (1− 2 𝑦 ) +2𝑐 𝑐 +𝑐 2 (1− 𝑦 2 ) + , , , , , s (y) = c + 2c c 1 y + 2c c + c 1 y + n n,0 n,1 n,0 n,2 n,0 n,1 2𝑐 𝑐 +𝑐 𝑐 (1− 𝑦 ) + , , , , 3/2 (128) 2[c c + c c ] 1 y + n,0 n,3 n,1 n,2 2𝑐 𝑐 +2𝑐 𝑐 +𝑐 (1− 𝑦 ) + ⋯ + , , , , , h i . (128) 2 2 ( ) ( ) 2𝑐 𝑐 1− 𝑦 +𝑐 1− 𝑦 2c c, + , 2c c + c 1 y + . . . + n,0 n,4 n,1 n,3 n,2 2n+1/2 2n+1 2 2 2 2c c 1 y + c 1 y n,2n n,2n+1 n,2n+1 7.4.3. Roots of Arccosine: Approximations for Even Powers of Arccosine and Arcsine The following theorem details a better approach for evaluating approximations for 2k 2k asin(y) and acos(y) , k 2 f1, 2, . . .g. Theorem 9. Root Based Approximation for Even Powers of Arccosine and Arcsine. 𝑎𝑠𝑖𝑛 AppliedMath 2023, 3 374 2k 2k Approximations of order n, for acos(y) and asin(y) , k 2 f1, 2, . . .g, respectively, are k k k k k k 2k p y y y y y y c (y) = (1 y) 1 1 1 1 1 1 (129) 2k,n 2k 2 r r r r r r 1 2 n 1 2 " # " # p p k k 2k 2 2 p 1 y 1 y s (y) = 1 1 y 1 1 2k,n 2k r r (130) " # " # " # " # p p p p k k k k 2 2 2 2 1 y 1 y 1 y 1 y 1 1 . . . 1 1 r r r r 2 n where r is the conjugate of r and r , r , . . . , r , r are the roots of the n th order approximation i 1 n n i 1 c (y) to acos(y) deﬁned in Corollary 1. Proof. Consider the nth order approximation c (y) to acos(y) deﬁned in Corollary 1. This approximation is denoted c and is of the form 2,n 2 2n+1 c (y) = c + c y + c y + . . . + c y (131) 2,n n,0 n,1 n,2 n,2n+1 This approximation can be written in the form p y y y y y y c (y) = (1 y) 1 1 ] 1 [1 ] [1 1 (132) 2,n 4 r r r r r r 1 2 n 1 2 It then follows that k k 2k p y y 2k k acos(y) c (y) = (1 y) 1 1 2k,n 2k 2 r r (133) k k k k y y y y 1 1 1 1 r r r r 2 n h i 2k The approximation, s (y), for asin(y) arises from the relationship asin(y) = acos 1 y . 2k,n 7.4.4. Approximations for Arccosine Squared The second order approximation for acos(y) is p y y y y c (y) = (1 y) 1 1 ] 1 [1 2,2 4 r r r r 21 22 (134) 1953 j4507 12, 833 j8339 r = + r = + 21 22 2500 2000 5000 5000 where j = 1. The relative error bound for this approximation, over the interval [0, 1], is 3.66 10 . The fourth and sixth order approximations are detailed in Appendix K and 6 8 have the respective relative error bounds of 2.48 10 and 2.25 10 . By using higher resolution in the approximations to the roots, slightly lower relative error bounds can be achieved. The stated root approximations represent a good compromise between accuracy and complexity. 7.4.5. Results 2k The relative error bounds associated with the nth order approximations for acos(y) 2k and asin(y) are detailed in Table 7. AppliedMath 2022, 2, FOR PEER REVIEW 30 AppliedMath 2023, 3 375 Table 7. Relative error bounds, over the interval [0, 1], for the approximations detailed in Theorem 9 for Tableacos(𝑦 7. Relative ) and error asin(𝑦 bounds, ) . over the interval [0, 1], for the approximations detailed in Theorem 9 2k 2k for acos(y) and asin(y) . Order, n, of Precision: Dig- Relative Error Relative Error Relative Error Order, n, of Precision: Relative Error Relative Error Relative Error Approx. its in Roots Bound: k = 1 Bound: k = 2 Bound: k = 3 Approx. Digits in Roots Bound: k = 1 Bound: k = 2 Bound: k = 3 −4 −4 −3 4 4 3 2 5 3.66 × 10 7.32 × 10 1.10 × 10 2 5 3.66 10 7.32 10 1.10 10 6 6 6 −6 −6 −6 4 4 8 8 2.48 × 2.48 10 10 4.96 × 4.96 10 10 7.43 × 7.43 10 10 8 8 8 6 9 2.25 10 4.49 10 6.74 10 −8 −8 −8 6 9 2.25 × 10 4.49 × 10 6.74 × 10 10 10 10 8 11 2.28 10 4.55 10 6.83 10 −10 −10 −10 8 11 2.28 × 10 4.55 × 10 6.83 × 10 12 12 12 10 13 2.93 10 5.85 10 8.78 10 −12 −12 −12 10 13 2.93 × 10 5.85 × 10 8.78 × 10 7.4.6. Comparison with Published Results 7.4.6. Comparison with Published Results Borwein [23] details approximations for even powers of arcsine and approximations Borwein [23] details approximations for even powers of arcsine and approximations for powers of two, four, six, eight and ten are detailed in Appendix L. The approximation for powers of two, four, six, eight and ten are detailed in Appendix L. The approximation for arcsine to the sixth power is for arcsine to the sixth power is " # n k1 m1 2k 2k 45 1 1 2 [k!] y 45 1 1 2 [𝑘!] 𝑦 asin(y) S (y) = (135) 6,n å å å 2 2 2 4 m p k 2k ! (𝑦) ≈𝑆 (𝑦) = ⋅ ⋅ ( ) (135) m=1 p=1 k=1 4 𝑚 𝑝 𝑘 (2𝑘)! As an example, the relative error in approximations for asin(y) , as deﬁned by S (y) 6,n As an example, the relative error in approximations for (𝑦) , as defined by (Equation (130)) and the Borwein approximation S y , are shown in Figure 15. The clear ( ) 6,n 𝑆 (𝑦) (Equation (130)) and the Borwein approximation 𝑆 (𝑦) , are shown in Figure 15. , , advantage of the root based approach over the series deﬁned by S (y) is evident. In 6,n The clear advantage of the root based approach over the series defined by 𝑆 (𝑦) is evi- particular, the root based approximations are sharp at the point one. dent. In particular, the root based approximations are sharp at the point one. Figure 15. Graph of the relative error in approximations to asin(y) , as deﬁned by S (y) for Figure 15. Graph of the relative error in approximations to 𝑎𝑠𝑖𝑛( 𝑦) , as defined by 𝑆 (𝑦) 6, for n 𝑛∈ {n 2 f3, 4, 5, 6, 7,}8, 9, 10g, along with root based approximations s (y) of orders 2, 3, 4, 5. 3,4,5,6,7,8,9,10 , along with root based approximations 𝑠 (𝑦) of 6,n orders 2,3,4,5. 7.5. Approximations for the Inverse Tangent Integral Function 7.5. Approximations for the Inverse Tangent Integral Function The inverse tangent integral function is deﬁned according to The inverse tangent integral function is defined according to atan((l) ) 𝑎𝑡𝑎𝑛 𝜆 T(y) = dl (136) 𝑇 (𝑦 ) = 𝑑𝜆 (136) and an explicit series form (e.g., Mathematica) is and an explicit series form (e.g., Mathematica) is ( ) [ ( ) ( )] 𝑇 𝑦 = ⋅ Li 𝑗 𝑦 −Li −𝑗 𝑦 , 𝑗 = √−1, T(y) = [Li (jy) Li (jy)], j = 1, 2 2 2𝑗 2j (137) (137) ( ) | | Li 𝑧 = , 𝑧 < 1, analytical continuation for |𝑧| > 1. Li (z) = , jzj< 1, analytical continuation for jzj >1. 𝑘 n k=1 The Taylor series for arctangent, as given by Equation (7), leads to the 𝑛𝑡ℎ order approx- imation, 𝑇 , for 𝑇 : 𝑎𝑠𝑖𝑛 𝑎𝑠𝑖𝑛 AppliedMath 2023, 3 376 AppliedMath 2022, 2, FOR PEER REVIEW 31 The Taylor series for arctangent, as given by Equation (7), leads to the nth order approxima- tion, T , for T: A,n 2k+1 (1) y (−1) ∙𝑦 T (y) = [u(y) u(y 1) å + A,n ( ) ( ) ( ) 𝑇 𝑦 =[𝑢 𝑦 −𝑢 𝑦− 1 ⋅ + (2k + 1) k= ( 0 ) 2𝑘 + 1 " # (138) (138) p (1) 𝜋 (−1) u y 1 ln y + ( ) ( ) å 𝑢 (𝑦− 1 ) ∙ln(𝑦 ) + 2k+1 (2k + 1) y 2 (2𝑘 k= + 0 1) 𝑦 where where𝑢 u is is the the unit unit step step fun function. ction. The r The relati elative er ve error ror inin appr approxim oximations, ations, o of or f orde ders rs one one to to ten, are shown in Figure 16. ten, are shown in Figure 16. Figure 16. Graph of the relative errors in the Taylor series (orders one to ten) based approximations Figure 16. Graph of the relative errors in the Taylor series (orders one to ten) based approximations for the for the inverse inversetangent integral, as given by tangent integral, as given by Equa Equation tion (138), (138), and and the the pr proposed approximations (or- oposed approximations (orders ders one to four) as specified in Equation (139). one to four) as speciﬁed in Equation (139). 7.5.1. Inverse Tangent Integral Approximation 7.5.1. Inverse Tangent Integral Approximation Based on the nth order approximation for arctangent, t , stated in Theorem 2, a nth Based on the 𝑛 th order approximation for arctangent, 𝑡 , stated in Theorem 2, a 𝑛 th order approximation to the inverse tangent integral is order approximation to the inverse tangent integral is 2n 2n 1 1 𝑇 (𝑦 )T=(y𝑑 ) = ∙ d 𝑑𝜆 = 𝑑 dl = 𝐼 (d 𝑦 ),𝑦 I (y)≥ , y0, 0, (139 (139) ) , å n,k , å n,k k ()/ (k+1)/2 [1 + 𝜆 ] 2 [1 + l ] k=0 k=0 where 𝑑 is defined in Theorem 2 and the integrals, 𝐼 ,𝐼 ,⋯,𝐼 are defined according where d is deﬁned in Theorem 2 and the integrals, I , I , , I are deﬁned according to 0 5 n,k 1 to I (y) = asinh(y) = ln y + 1 + y , I (y) = atan(y), (140) 𝐼 (𝑦 ) =asinh(𝑦 ) =ln 𝑦 + 1+𝑦 , 𝐼 (𝑦 ) =atan(𝑦 ), (140) 0 1 𝑦 𝑦 atan(𝑦) ( ) ( ) 𝐼 𝑦 = , 𝐼 𝑦 = + (141) y y atan(y) 2(1 + 𝑦 ) 2 1+𝑦 I (y) = p , I (y) = + (141) 2 3 2(1 + y ) 2 1 + y 𝑦(1 + 2𝑦 /3) 5𝑦(1 + 3𝑦 /5) 3atan(𝑦) ( ) 2 ( ) 2 (142) 𝐼 𝑦 = , 𝐼 𝑦 = + y 1 / + 2y /3 5y 1 + 3y /5 3atan(y) (1 + 𝑦 ) 8(1 + 𝑦 ) 8 I (y) = , I (y) = + (142) 4 5 3/2 2 2 8 8 1 + y (1 + y ) ( ) The first order approximation, for the inverse arctangent integral, is The ﬁrst order approximation, for the inverse arctangent integral, is 𝜋 3𝜋 3𝜋 3𝜋 𝑦 ( ) ( ) 𝑇 𝑦 = ∙ln 𝑦 + 1+𝑦 +−2−2𝜋 + atan 𝑦 +[3 + − ]∙ (143) 2 2 2 4 2 4 p 3p 3p 3p 1+𝑦 y T (y) = ln y + 1 + y + 2 2p + atan(y) + 3 + p (143) 2 4 2 4 1 + y Second and third order approximations are detailed in Appendix M. Second and third order approximations are detailed in Appendix M. 7.5.2. Notes and Relative Error 7.5.2. Notes and Relative Error The approximations, 𝑇 , 𝑛∈ {1,2, … } , are valid over the positive real line and the rel- ative e The rror appr in the oximations, approximations, of o T , n 2 f1, 2, rders . . .g,one to four are valid over , are sho the positive wn in Figure real line 16. and As the is eviden relative t in th error is F in igu the re, the approxim approximations, atioof ns or ha ders ve a one lowe to r re four lati,ve err are shown or bound in tha Figur n the e 16 d . is As - jointly is evident define in d Taylor this Figur ser e, iethe s approx approximations imations dehave fined b a y lower Equation relative (138). The re error bound lative error than the bounds associated with the approximations are detailed in Table 8. AppliedMath 2023, 3 377 disjointly deﬁned Taylor series approximations deﬁned by Equation (138). The relative error bounds associated with the approximations are detailed in Table 8. Table 8. Relative error bounds, over the interval [0,¥), for Taylor series based approximation, and the approximations speciﬁed in Equation (139), for the inverse tangent integral function. Relative Error Bound: Order of Approx. n Relative Error Bound: T Taylor Series T A,n 2 3 1 2.96 10 4.78 10 2 4 2 1.41 10 2.88 10 3 5 3 8.17 10 2.23 10 3 6 4 5.31 10 1.95 10 3 7 5 3.72 10 1.83 10 3 8 6 2.74 10 1.80 10 7.5.3. Approximation of Catalan’s Constant As Catalan’s constant can be deﬁned according to atan l ( ) G = dl (144) it follows that approximations for this constant, of orders two and four, can be deﬁned according to p 35 86 2 20 61 1 55 15p G = ln[1 + 2] + + p p + p + p (145) 2 6 9 3 8 16 4 2 8 2 p 298, 369 2, 609, 456 2 10, 342 218, 147 G = ln[1 + 2] + + p p + 2 630 3675 21 224 2 (146) 557 14, 529 3465p p + 16 64 32 2 4 6 The respective relative errors in these approximation are 2.25 10 and 1.03 10 . 7.6. Approximations for Unknown Integrals The different forms for the approximations for arcsine, arccosine and arctangent, potentially, can lead to approximations for unknown integrals involving these functions. Four examples are detailed below. 7.6.1. Example 1 t 2 The function 4acos e / is an approximation to the unit step function for y 0 after a transient rise time. Using the approximation form, c , detailed in Corollary 1 for arccosine, the approximation to the integral of this function (scaled by p /4 ) can be deﬁned: h i 2 2n+1 2 p y n,k t ky acos e dt I y = + 1 e , y > 0. (147) ( ) n å 4 k k=1 AppliedMath 2023, 3 378 The third order approximation is 2 2 2y 3y 139 271p 319p p y e pe I (y) = + + pe + + 300 630 1680 4 2 18 2 2 979 31p 35p 944 46p 21p 4y 5y + e + + e + (148) 180 6 16 75 5 5 2 2 48 55p 35p 112 61p 5p 6y 7y + e + + e 5 9 12 45 42 7 and the relative error bound associated with this approximation, over the interval [0, ¥) , is 2.32 10 . 7.6.2. Example 2 Using the approximation form, t , detailed in Corollary 1 for arctangent, the following approximation can be deﬁned 2 3 ky h i 2n+1 p 2 p y 2c n,k 4 5 atan e 1 dt I (y) = + 1 e , y > 0. (149) n å 4 k k=1 Mathematica, for example, speciﬁes this integral in terms of the poly-logarithmic function. The third order approximation is 2 2 3y/2 139 271p 319p p y pe y/2 y I (y) = + + 2pe e + + 150 315 840 4 9 2 2 979 31p 35p 1888 92p 42p 2y 5y/2 (150) + e + + e + 90 3 8 75 5 5 2 2 96 110p 35p 224 61p 10p 3y 7y/2 + e + + e 5 9 6 45 21 7 and the relative error bound associated with this approximation, over the interval 0, ¥ , [ ) is 2.32 10 . 7.6.3. Example 3 The following integral does not have an explicit analytical form but the approxima- tions, t , detailed in Corollary 1, leads to "s # Z Z 2t 2n+1 2n+1 y y kt ky atan 1 dt I (y) = c (1 + t) e dt = p (y)e , (151) n å n,k å k 0 0 (1 + t) k=0 k=0 y > 0, where the polynomials p , . . . , p can readily be established. For the case of 0 2n+1 n = 2, the relative error bound, associated the interval [0, ¥) , is 3.00 10 . 7.6.4. Example 4 Consider the deﬁnite integral deﬁned by Sofo and Nimbran [27] (example 2.8, factor of 1/4 missing): " # i+1 n i 1 (1) 1 2 2 I(1) = t ln (t) atan(t) dt I = (152) S,n å å 4 2k 1 i(i + 1) i=1 k=1 AppliedMath 2023, 3 379 The polynomial approximation, t , for arctangent detailed in Theorem 5 and for the interval 0 y 1, yields R 2n+1 2n+1 R y y 2 2 i+k+1 2 I y = tln t) atan t) dt I y = d d t ln t) dt ( ) ( ) å å n n,i n,k 0 0 i=1 k=1 2 3 (153) 1 i + k + 2 ln y + [ ] ( ) i+k+2 2n+1 2n+1 2y 2 2 4 5 = d d i + k å å n,i n,k (i + k + 2) 2 + 2i + 2k + ik + ln y) i=1 k=1 for 0 < y 1. For the case of y = 1 the approximation is 2n+1 2n+1 I (1) = d d (154) n å å n,i n,k (i + k + 2) i=1 k=1 The relative errors in the approximations I and I 1 are detailed in Table 9. The relative ( ) S,n n errors in the approximations I (1), n 2 f1, 2, . . . ,6} are shown in Figure 17. From the results shown in Table 9, it is clear that the approximations speciﬁed by Equation (154) converge signiﬁcantly faster than the approximations detailed by Sofo and Nimbran [27] (Equation (152)). In addition, the approximation, t , for arctangent, underpins the more general approximation, as speciﬁed by Equation (153), for the integral I (y), 0 < y 1. Table 9. Table of the relative errors associated with the approximations I and I (y) as deﬁned by S,n Equations (152) and (154). Order of Relative Error in Relative Error in Relative Error Bound Approx: n Approx: I Approx. I (1) for I (y), 0<y1 n n S,2n+1 AppliedMath 2022, 2, FOR PEER REVIEW 34 2 2 2 1 2.15 10 3.16 10 4.04 10 3 3 3 2 5.44 10 2.24 10 2.96 10 3 5 5 3 1.97 10 3.18 10 3.34 10 4 5 5 results shown in Table 9, it is clear that the approximations specified by Equation (154) 4 8.85 10 4.16 10 5.01 10 4 7 7 converge s 6 ignificantly fas 2.59 ter th 10 an the approxima6.82 tion s det 10 ailed by Sofo and Ni 8.84mbran 10 [27] 4 8 8 1.02 10 1.84 10 2.30 10 (Equation (152)). In addition, the approximation, 𝑡 , for arctangent, underpins the more 5 10 10 10 4.82 10 3.48 10 4.58 10 general approximation, as specified by Equation (153), for the integral I (𝑦) , 0<𝑦 ≤ 1 . Figure 17. Graph of the relative errors in the approximations, of orders one to six, as deﬁned by I (y) Figure 17. Graph of the relative errors in the approximations, of orders one to six, as defined by n 𝐼 (Equation (𝑦) (Equation (153)).(153)). 8. Summary and Conclusions Table 9. Table of the relative errors associated with the approximations 𝐼 and 𝐼 (𝑦 ) as defined 8.1. Summary of Results by Equations (152) and (154). The approximations detailed in the paper for arcsine and arctangent are tabulated, rOrder of Ap- espectively, in Tables Relative Error 10 and 11 in Ap- . Relative Error in Ap- Relative Error Bound for prox: n prox: 𝑰 prox. 𝑰 (𝟏) 𝑰 (𝒚) , 0<𝒚≤𝟏 𝑺 ,𝟐𝒏𝟏 𝒏 𝒏 1 2.15 × 10 3.16 × 10 4.04 × 10 2 5.44 × 10 2.24 × 10 2.96 × 10 3 1.97 × 10 3.18 × 10 3.34 × 10 4 8.85 × 10 4.16 × 10 5.01 × 10 6 2.59 × 10 6.82 × 10 8.84 × 10 8 1.02 × 10 1.84 × 10 2.30 × 10 10 4.82 × 10 3.48 × 10 4.58 × 10 8. Summary and Conclusions 8.1. Summary of Results The approximations detailed in the paper for arcsine and arctangent are tabulated, respectively, in Tables 10 and 11. Table 10. Approximations for arcsine. The coefficients 𝑐 , 𝑑 and 𝛿 are defined in the associ- . . . ated reference. Relative Error Bound for Reference Approximation for Arcsine of Order n [0, 1], n = 4 𝜋 2.49 × 10 Corollary 1 ( ) − 𝑐 𝑦 , 𝑐 1− 𝑦 , , 1.24 × 10 1.78 × 10 Theorem 2 − 1− 𝑦 𝑑 𝑦 , 𝑦 𝑑 (1− 𝑦 ) , , 4.72 × 10 ( ) 1 1− 1− 𝑦 1 − 1− 𝑦 + 𝑑 ⋅ Theorem 3 1.00 × 10 𝑦 2+ 𝑘 AppliedMath 2023, 3 380 Table 10. Approximations for arcsine. The coefﬁcients c , d and d are deﬁned in the associated n.k n.k n.k reference. Reference Approximation for Arcsine of Order n Relative Error Bound for [0, 1], n = 4 s s 2.49 10 2n+1 2n+1 Corollary 1 k/2 k 2 6 c y , c 1 y å å ( ) 1.24 10 n,k n,k k=0 k=0 1.78 10 p 2n 2n p k/2 k 2 Theorem 2 2 1 y å d y , y å d 1 y n,k n,k 4.72 10 k=0 k=0 " # 1+k/2 p 2n 6 1 1 y Theorem 3 1.00 10 1 1 y + d n,k y 2 + k k=0 2 2n+1 d y d y d y n,1 n,2 n,2n+1 p p + + . . . + , 0 y Theorem 5 2 5 n+1/2 1 y 2 3.34 10 1 y 2 1 y ( ) (Equation (86)) n+1/2 2 2 d 1 y d 1 y p d 1 y n,1 n,2 n,2n+1 . . . 2 2n+1 2 y y y 1/ 2 < y 1 Table 11. Approximations for arctangent. The coefﬁcients c , d and d are deﬁned in the n,k n,k n,k associated reference. Reference Approximation for Arctangent of Order n Relative Error Bound for [0,¥), n=4 1.24 10 2n+1 u2n+1 c c y Corollary 1 p n,k n,k , 2.49 10 å å k/2 k/2 2 2 2 k=0 k=0 (1 + y ) (1 + y ) 4.72 10 2n 2n d d y n,k n,k Theorem 2 y å , å 1.78 10 (k+1)/2 (k+1)/2 2 2 2 k=0 k=0 (1 + y ) (1 + y ) 2 2 33 Theorem 3 6 2n 6 77 1.00 10 1 + y 1 d 1 n,k 6 6 77 1 p + 1 4 4 55 y 2 + k k 1 + y k=0 1+ (1 + y ) 2 2n+1 d y + d y + . . . + d y , 0 y 1 n,1 n,2 n,2n+1 p d d d 5 n,1 n,2 n,2n+1 Theorem 5 3.34 10 . . . , 1 < y < ¥ 2 2n+1 2 y y y 2 8 3 4 2n+3 d y d y > d y n,1 n,2 n,2n+1 + + . . . + 0 y 1 6 > 3 4 2n + 3 6 > 6 Theorem 6 6.34 10 6 > 6 > d d p y 1 n,1 n,2n+1 + . . . + + d (y 1) 2 y n,1 + 3 2n + 3 4 1 + y 2 > 6 > > d 1 1 n,2n+1 6 > d ln(y) d 1 . . . 1 6 > n,2 n,3 > 2n1 > y 2n 1 y 1 < y < ¥ For arcsine, the approximation form, s detailed in Theorem 2, can be written in the simple form s (y) = y p (y) + p (y) 1 y (155) 1 2 where p and p are polynomial functions. The approximation s , detailed in Theorem 3, 1 2 n has the lowest relative error bound for a set order (e.g., order four). 8.2. Conclusions Based on the geometry of a radial function, and the use of a two point spline approxi- mation, approximations of arbitrary accuracy, for arcsine, arccosine and arctangent, can be speciﬁed. Explicit expressions for the coefﬁcients used in the approximations were detailed and convergence was proved. The approximations for arcsine and arccosine are sharp at AppliedMath 2023, 3 381 the point zero and one and have a deﬁned relative error bound for the interval 0, 1 . Alter- [ ] native approximations were established based on a known integration result and a known differentiation result. The approximations have the forms detailed in Tables 10 and 11. By utilizing the anti-symmetric relationship for arctangent around the point one, a two point spline approximation was used to establish approximations for this function as well as for arcsine and arccosine. Alternative approximations were established by using a known integral result. Iteration utilizing the Newton-Raphson method, and based on any of the proposed approximations, yields results with signiﬁcantly higher accuracy. The approximations exhibit quadratic convergence with iteration. Applications of the approximations include: ﬁrst, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, it was shown how to use upper and lower bounded approximations to deﬁne approximations with signiﬁcantly higher accuracy. Third, it was shown that the approximation s , detailed in Corollary 1, leads to a simple approximation form for the square of arcsine which has better conver- gence than established series for this function. By utilizing the roots of the square of the approximations to arccosine detailed in Corollary 1, it was shown how approximations to arccosine and arcsine, to even power orders, can be established. It was shown that the relative error bounds associated with such approximations are signiﬁcantly lower that published approximations. Fourth, approximations for the inverse tangent integral function were proposed which have signiﬁcantly lower relative error bounds over the inter- val [0,¥), than established Taylor series based approximations. Fifth, the approximation forms for arccosine and arctangent were utilized to establish approximations to several unknown integrals. Funding: This research received no external funding. Institutional Review Board Statement: Not relevant. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The author declares no conﬂict of interest. Appendix A. Approximations Based on Angle Subdivision Given the coordinate (x, y) of a point on the ﬁrst quadrant of the unit circle, and the corresponding angle q, as deﬁned by q = acos(x) and q = asin(y), the following deﬁnitions can be made: q q s = sin , c = cos , i 2 f0, 1, . . .g. (A1) i i i i 2 2 Algorithms for determining s and c arise from half-angle formulas and are: i i p p 1 1 p p s = 1 c , c = 1 + c , i 2 f1, 2, . . .g i i1 i i1 2 2 (A2) p p 2 2 s = y = 1 x = sin(q), c = x = 1 y = cos(q) 0 0 The following result can be proved, following the approach detailed in [15] (Section 6.4 and Appendix I). Theorem A1. Approximation for Arcsine and Arccosine. Approximations for asin(y) and acos(x), of order n, are: 2 3 h i (1) p k, s 1 y 6 7 i k+1 asin y s y = 2 d s 1 y p k, 0 + (A3) ( ) ( ) 4 [ ] 5 i,n å n,k h i p (2k+1)/2 2 2 k=0 1 s 1 y i AppliedMath 2023, 3 382 2 3 h i n h i (1) p k, 1 c (x) (k+1)/2 6 7 i 2 acos(x) c (x) = 2 d 1 c (x) p[k, 0] + (A4) 4 5 i,n å n,k 2k+1 c (x) k=0 where p k, t = 1 t p k 1, t + 2k 1 t p k, t , p 0, t = 1 (A5) ( ) ( ) ( ) ( ) ( ) dt n! (2n + 1 k)! d = (A6) n,k (n k)!(k + 1)! 2 (2n + 1)! Proof. The angle q/2 can be deﬁned according to the standard path length formula along the unit circle from the point (0, 1) to the point (s , c )(the point consistent with the angle i i p/2 q/2 ): Z Z 2 1c q 1 i 1 = p dl = p dl, i e f1, 2, . . .g. (A7) 2 2 1 l 1 l The integral can be approximated by using the general integral approximation [15] (eqn. 14): n h i k+1 k (k) (k) f (l)dl d (t a) f (a) + (1) f (t) (A8) å n, k k=0 where for the case being considered 1 p(k, t) (k) f t = p , f t = , k e 0, 1, . . . . (A9) ( ) ( ) f g (2k+1)/2 1 t [1 t ] Here, p(k, t) is speciﬁed by Equation (A5). For the case of a = 0 and t = s or t = 1 c , Equation (A8), respectively, leads to the required results: q = asin(y) s (y) i, n 2 3 h i (1) p k, s 1 y n p i (A10) 6 7 k+1 i 2 = 2 d s 1 y p[k, 0] + å 4 5 n,k h i p (2k+1)/2 k=0 2 2 1 s 1 y q = acos(x) c (x) i,n 2 3 h q i (1) p k, 1 c (x) i (A11) (k+1)/2 6 7 i 2 = 2 d 1 c (x) p[k, 0] + å 4 5 n,k i 2k+1 c (x) k=0 Explicit Approximations for Arccosine Some examples of the approximations for arccosine, as speciﬁed by Equation (A4), are detailed below: First, based on q/2, the second order spline approximation yields p p h i 121 1 x x 1 x 13 19x 37x c (x) = p 1 + + + (A12) 1,2 5/2 121 15 10 30 120 2 (1 + x) AppliedMath 2023, 3 383 which has a relative error bound, for the interval 0, 1 of 5.56 10 . Second, based on [ ] q/4, the second order spline approximation yields p " # 121 2 2 1 + x 1 + x c x = 1 p + ( ) 2,2 121 2 (A13) p p p p 2 2 1 + x 178 74x 38 2 1 + x [ + + 5/2 15 15 5 [2 + 2 1 + x] which has a relative error bound, for the interval [0, 1], of 1.71 10 . Third, based on q/16, the ﬁrst order spline approximation yields 2 3 " # p p 6 7 10 1 + 2 + 2 + 2 1 + x 6 7 p 10 6 7 c x = 4 2 2 + 2 + 2 1 + x 1 + (A14) ( ) 6 7 4,1 " # 3/2 6 q 7 p p 4 5 3 2 + 2 + 2 + 2 1 + x which has a relative error bound, for the interval [0, 1], of 1.19 10 . Appendix B. Explicit Approximations for Radial Function Approximations for r , as speciﬁed by Theorem 1 and of orders one to six, are detailed below with the coefﬁcients C , k 2 f0, 1, . . . , 2n + 1g, being speciﬁed in Table A1: n,k 2 3 f (y) = py + C y + C y (A15) 1 1,2 1,3 2 3 4 5 f (y) = py + 2y + C y + C y + C y (A16) 2 2,3 2,4 2,5 2 3 p py 2 4 5 6 7 f (y) = py + 2y + C y + C y + C y + C y (A17) 3 3,5 3,6 3,7 3,4 4 6 2 3 4 p py y 2 5 6 7 8 9 f (y) = py + 2y + + C y + C y + C y + C y + C y (A18) 4 4,5 4,6 4,7 4,8 4,9 4 6 3 2 3 4 5 p py y 3py 2 6 7 8 f (y) = py + 2y + + C y + C y + C y + 5 5,6 5,7 5,8 4 6 3 40 (A19) 9 10 11 C y + C y + C y 5,9 5,10 5,11 2 3 4 5 6 p py y 3py 8y 2 7 8 9 f (y) = py + 2y + + + C y + C y + C y + 6 6,7 6,8 6,9 4 6 3 40 45 (A20) 10 11 12 13 C y + C y + C y + C y 6,10 6,11 6,12 6,13 AppliedMath 2023, 3 384 Table A1. Table of coefﬁcients. The lower order coefﬁcients that are not listed are deﬁned according to C = C , k 2 f0, 1, . . . , n 1g. n,k n1,k Order of Approx. Coefﬁcients 2 2 p p C = , C = 1 0,0 0,1 4 4 2 2 3p p C = p, C = 3 + 2p , C = 2 p + 1,1 1,2 1,3 4 2 16 5p 2 C = 2, C = + 6p 2,2 2,3 3 2 2 2 35 15p 16 3p C = 8p + , C = + 3p 2,4 2,5 3 4 3 2 p 979 62p 35p 944 C = , C = + , C = 46p + 21p 3,3 3,5 3,4 6 45 3 4 15 288 110p 35p 784 61p C = + , C = + 5p 3,6 3,7 5 3 2 45 6 1 8704 145p 63p C = , C = + 4,4 4,5 3 105 2 2 19, 624 692p 45, 056 575p 2 2 C = + 105p , C = + 135p 4,6 4,7 63 3 105 2 2 2 27, 508 315p 18, 944 215p 35p C = 164p + , C = + 4,8 4,9 105 4 315 6 2 3p 166, 792 15, 707p 231p 66, 304 8689p C = , C = + , C = + 495p 5,5 5,6 5,7 40 525 60 2 45 8 854, 948 3715p 3465p 87, 552 38, 947p C = + , C = + 770p 5,8 5,9 315 2 4 35 24 364, 288 14, 409p 693p 338, 176 5183p C = + , C = + 63p 5,10 5,11 315 20 2 1575 40 8 63, 125, 504 9611p C = , C = + 429p 6,6 6,7 45 51, 975 10 116, 868, 932 24, 642p 9009p 6, 002, 688 43, 043p C = + , C = + 5005p 6,8 6,9 17, 325 5 4 385 4 200, 238, 464 63, 684p 46, 544, 896 2 2 C = + 6006p , C = + 8589p 4095p 6,10 6,11 10, 395 5 3465 86, 876, 288 46, 814p 3003p 40, 687, 616 19, 061p C = + , C = + 231p 6,12 6,13 17325 15 2 51, 975 40 Appendix C. Explicit Approximations for Arccosine Explicit approximations for arccosine, of orders three to six and arising from Corollary 1, are: 2 3 p py 2 4 5 6 7 c (y) = py + y + c y + c y + c y + c y (A21) 3 3,4 3,5 3,6 3,7 4 6 2 3 4 p py y 2 5 6 7 8 9 (A22) c (y) = py + y + + c y + c y + c y + c y + c y 4 4,5 4,6 4,7 4,8 4,9 4 6 3 2 3 4 5 p py y 3py 2 6 7 8 py + y + + c y + c y + c y 5,6 5,7 5,8 c (y) = 4 6 3 40 (A23) 9 10 11 +c y + c y + c y 5,9 5,10 5,11 AppliedMath 2023, 3 385 2 3 4 5 6 p py y 3py 8y 2 7 8 py + y + + + c y + c y 6,7 6,8 4 6 3 40 45 c (y) = (A24) 9 10 11 12 13 +c y + c y + c y + c y + c y 6,9 6,10 6,11 6,12 6,13 Appendix D. Approximations for Arcsine of Orders Three to Four Approximations for arcsine, of orders three and four and arising from Theorem 2, are: p p py 3 4 5 6 s (y) = 1 y y + + d y + d y + d y + d y 3 3,3 3,5 3,6 3,4 2 2 4 2 3 p 1 y (A25) p 3/2 2 2 2 2 1 y + + d 1 y + d 1 y + 3,3 3,4 6 7 2 4 s (y) = y 4 5 5/2 3 2 2 d 1 y + d 1 y 3,5 3,6 2 2 1958 124p 35p 472 105p d = + , d = + 115p , 3,3 3,4 45 3 2 3 2 (A26) 2 2 864 105p 2744 427p 35p d = 110p + , d = + . 3,5 3,6 5 2 45 12 2 2 3 2 3 p py 2y 4 5 6 y + + d y + d y + d y + 2 4,4 4,5 4,6 4 5 s (y) = 1 y 4 2 4 3 7 8 d y + d y 4,7 4,8 2 3 3/2 2 2 (A27) p 1 y 2 1 y 6 7 1 y + + d 1 y + 4,4 6 7 A 2 4 3 s (y) = y6 7 4 5 5/2 3 7/2 4 2 2 2 2 d 1 y + d 1 y + d 1 y + d 1 y 4,5 4,6 4,7 4,8 4352 725p 315p 19, 624 d = + , d = + 692p 315p , 4,4 4,5 21 4 4 21 22, 528 4025p 945p 110, 032 (A28) d = + , d = + 656p 315p , 4,6 4,7 15 4 2 105 9472 645p 315p d = + . 4,8 35 4 4 Appendix E. Proof of Theorem 4 Consider the differential equation stated in Equation (68): h i h i (1) (1) 2 2 1 y f (y) + # (y) 2y 4 f (y) + # (y) y = 0 (A29) n n n n and the nth order approximation, f , detailed in Theorem 1: f (y) = C + C y + . . . + n n n,0 n,1 2n+1 C y . As # 0 = 0, the following form for the error function is assumed: ( ) n,2n+1 n 2 3 # (y) = [k C ]y+[k C + 1]y + [k C ]y + . . . + n n,2 n,2 n,3 n,3 n,1 n,1 (A30) 2n+1 2n+2 [k C ]y + k y + . . . n,2n+1 n,2n+1 n,2n+2 with unknown coefﬁcients k , k , . Use of this form in Equation (A29) leads to n,1 n,2 " # 2n k + 2k y + . . . + (2n + 1)k y + n,1 n,2 n,2n+1 1 y 2n+1 (A31) (2n + 2)k y + . . . n,2n+2 2 2n+1 2n+2 4 C + k y + k y + . . . + k y + k y + . . . = 0 n,0 n,1 n,2 n,2n+1 n,2n+2 AppliedMath 2023, 3 386 i.e., ¥ ¥ ¥ 2 i+j2 i 1 y i jk k y 4C 4 k y = 0 (A32) å å n,i n,j n,0 å n,i i=1 j=1 i=1 As C = p /4, n e 0, 1, 2, , it follows that the coefﬁcients k , i e 1, 2, , are f g f g n,o n,i independent of n, leading to ¥ ¥ ¥ ¥ ¥ i+j2 i+j i i jk k y i jk k y 4C 4 k y = 0 (A33) n,0 å å i j å å i j å i i=1 j=1 i=1 j=1 i=1 0 2 By sequentially considering the coefﬁcients of y , y, y , the constants k , i e f1, 2,g, 0 2 can be determined. First, the coefﬁcient of y yields k = 4 C , leading to k = 2 C = n,0 1 n,0 p. The negative solution is required as # (y) = [k C ]y + . . . and C = p . Second, 1 n,1 n,1 the coefﬁcient of y yields 4k k 4k = 0 , leading to k = 1. Third, the coefﬁcient of y 1 2 1 2 2 2 yields 6k k + 4k k 4k = 0, leading to k = k /6 = p/6. For the general case, the 1 3 2 3 1 2 1 q1 coefﬁcient of y , q 3, yields i jk k i jk k 4k = 0 (A34) i j i j q1 å å i,j e f1,2,g, i+j=q+1 i,j e f1,2,g, i+j=q1 Thus: (1q)k k + 2(q 1)k k + . . . + (q 1)(2)k k + (q1)k k 1 q 2 q1 q1 2 q 1 (A35) 1(q 2)k k + 2(q 3)k k + . . . + (q 2)(1)k k 4k = 0 1 q2 2 q3 q2 1 q1 i.e., q1 q2 2qk k + u q u + 1 k k u q u 1 k k 4k = 0 (A36) ( ) ( ) 1 q u qu+1 u qu1 q1 å å u=2 u=1 leading to q1 q2 4k u(q + 1 u)k k + u(q u 1)k k å u å u q1 q+1u qu1 u=2 u=1 k = (A37) 2qk for q 2 f3, 4, . . .g. Coefﬁcient Values Use of Equation (A37), for q 3, leads to the following list of coefﬁcient values: p 1 k = p, k = 1, k = , k = , 1 2 3 4 6 3 3p 8 5p 4 (A38) k = , k = , k = , k = , 5 6 7 8 40 45 112 35 35p 128 63p 128 k = , k = , k = , k = , 9 10 11 12 1152 1575 2816 2079 and the values are consistent with the result k = C , for i 2 f1, 3, 4, . . .g (see Table A1 i i,i for C , C , . . . , C ). It is the case that k = C 1. These results are consistent, see 1,1 3,3 6,6 2 2,2 (i) (i) Equation (A30), with the requirement that f (0) = f (0), i 2 f0, 1, . . . , ng which implies (i) # (0) = 0, i 2 f0, 1, . . . , ng. n AppliedMath 2023, 3 387 From Equation (A30), the result C = C , i 2 1, 2, . . . , n then follows and, for f g n,i i,i n 2 f3, 4, . . .g, it is the case that 2n+1 ¥ 2n+1 ¥ i i i i # (y) = [k C ]y + k y = [C C ]y + C y (A39) n å i n,i å i å i,i n,i å i,i i=n+1 i=2n+2 i=n+1 i=2n+2 which is the required result. Appendix F. Proof of Theorem 1 Consider the form for the nth order two point spline approximation, denoted f , to a function f as detailed in [15] (eqn. 40), and the alternative form given in [16] (eqn. 70). Based on the points zero and one, the nth order approximation is n n n+1 r r n+1 f (y) = (1 y) a y + y b (1 y) , (A40) n å n,r å n,r r=0 r=0 ru r (ru) r (ru) f (0) (n + u)! (1) f (1) (n + u)! a = , b = , (A41) n,r n,r å å (r u)! u!n! (r u)! u!n! u=0 u=0 2 2 2 r 2 0, 1, . . . , n , where f y = r y , r y = acos(y) + y . f g ( ) ( ) ( ) (k) (k) The sequence of numbers deﬁned by f (0) and f (1), for k 2 f0, 1, 2, . . .g, respec- tively, are: ,p, 4,p, 8,9p, 128,225p, 4608,11, 025p, 294, 912, . . . (A42) 8 8 24 128 640 7680 3584 229, 376 18, 579, 456 1, 0, , , , , , , , , , . . . (A43) 3 15 35 105 231 1001 143 2431 46, 189 For the ﬁrst sequence, the ratios of the ﬁfth to the third term, the seventh to the ﬁfth term,...are: 9 225 11, 025 2 2 2 = 3 , = 5 , = 49 = 7 , (A44) 1 9 225 The ratios of the sixth to the fourth term, eight to the sixth term, . . . are: 128 4608 294, 912 2 2 2 2 2 2 = 16 = 2 2 , = 36 = 2 3 , = 64 = 2 4 , (A45) 8 128 4608 (k) It then follows that the general iteration formula for f (0) is: (1) (3) (2) (4) f (0) = , f (0) = f (0) = p, f (0) = 4, f (0) = 8, (A46) (k) (k2) f (0) = (k 2) f (0), k 2 f5, 6, 7, . . .g. (k) (k) (k1) The general iteration form for f (1) arises by considering the ratios f (1)/ f (1), for k 2 f5, 6, 7, . . .g, leading to: (1) (2) (3) f (1) = 1, f (1) = 0, f (1) = 8/3, f (1) = 8/15, k 2 (A47) (1) (k 1) (k) (k1) f (1) = f (1) , k 2 f4, 5, 6, . . .g. 2k 1 AppliedMath 2023, 3 388 Appendix F.1. Formula for Coefﬁcients in Standard Polynomial Form The goal is to write the approximation f , as deﬁned by Equation (A40), in the form 2n+1 f (y) = C y (A48) n å n,K k=0 To this end, the binomial formula (1) i! (1 y) = y (A49) (i k)!k! k=0 mplies n+1 (1) (n + 1)! n k f (y) = [a + a y + . . . + a y ] y + n n,0 n,1 n,n (n + 1 k)!k! k=0 2 3 (1) r! (A50) b + b (1 y) + . . . + b y + . . . + 6 n,r å 7 n,0 n,1 6 (r k)!k! 7 k=0 n+1 6 7 6 7 4 5 (1) n! b y n,n (n k)!k! k=0 Thus: k k 1) (n + 1)! 1) (n + 1)! n+1 n+1 k k+1 f (y) = a å y + a å y + . . . + n n,0 n,1 n + 1 k !k! n + 1 k !k! ( ) ( ) k=0 k=0 k k 1) (n + 1)! 1) (n + 1)! n+1 n+1 k+r k+n a å y + . . . + a å y + n,r n,n (n + 1 k)!k! (n + 1 k)!k! k=0 k=0 (A51) 1) r! n+1 n+1 n+2 n+k+1 b y + b y y + . . . + b y + . . . + n,0 n,r å n,1 (r k)!k! k=0 1) n! n+k+1 b y n,n å (n k)!k! k=0 n+1 (1) (n + 1)! i k+r For 0 i n, y is associated with the value of k in the summation a å y n,r (n + 1 k)!k! k=0 which is such that k + r = i, k 0, i.e., k = i r and 0 r i. Thus: ir (1) (n + 1)! C = a , 0 i n. (A52) n,i n,r (n + 1 + r i)!(i r)! r=0 For n + 1 i 2n + 1, the lowest value of r, such that there is a term associated with y in a , satisﬁes the constraint n + 1 + r = i, i.e., r = i n 1. The term y is also associated n,r with the index n + k + 1 = i, k 0, in b , i.e., k = i n 1, and with the lowest value of r n,r being consistent with n + r + 1 = i. Thus: ir (1) (n + 1)! C = å a + n,r n,i n + 1 + r i ! i r ! ( ) ( ) r=in1 (A53) in1 (1) r! b , n + 1 i 2n + 1. n,r (r + n + 1 i)!(i n 1)! r=in1 AppliedMath 2023, 3 389 Appendix F.2. Nature of Coefﬁcients Consider a and C as deﬁned by Equations (A41) and (A52), whereupon it fol- n,r n,i lows that (1) a = f (0), a = (n + 1) f (0) + f (0), (A54) n,0 n,1 (1) C = a = f 0 , C = n + 1 a + a = f 0 . (A55) ( ) ( ) ( ) n,0 n,0 n,1 n,0 n,1 It can readily be shown that (i) f (0) C = , i 2 f0, 1, . . . , ng. (A56) n,i i! (i) (i) This result is consistent with the requirement, f (0) = f (0) for i 2 f0, 1, . . . , ng, associated with a two point spline approximation of order n. Appendix G. Third and Fourth Order Approximations for Arctangent Approximations for arctangent, of orders three and four and arising from Theorem 5, are: y 55 35p 265 331 35p 63 4 5 6 7 > y y + 21p y y + 5p y , 3 2 4 4 6 2 4 > 0 y 1 t (y) = (A57) p 1 1 55 35p 1 265 1 331 35p 1 > + + 21p + 3 4 5 6 2 y 3y 2 4 y 4 y 6 2 y > 63 1 : 5p , y > 1 4 y y 395 63p 1979 1697 5 6 7 > y y + 105p y 135p y + 3 4 2 6 4 495 315p 35p 8 9 > y 55 y , 0 y 1 2 4 2 t y = (A58) ( ) p 1 1 395 63p 1 1979 1 > + + 105p + 3 5 6 2 y 3y 4 2 y 6 y 1697 1 495 315p 1 35p 1 : 135p + 55 , y > 1 7 8 9 4 y 2 4 y 2 y Appendix H. Alternative Third and Fourth Order Approximations for Arctangent Third and fourth order approximations for arctangent, and arising from Theorem 6, are: 8 2 3 3 5 2y 2y 55 35p 265 6 7 > y + y + 6p y 6 7 3 15 6 12 14 6 7 6 7 0 y 1 6 7 > 2 331 35p 7 10p > 1 + y 8 9 > 4 5 > y + y > 24 8 2 9 2 3 t (y) = (A59) 6121 131p 63 1 331 35p 1 + 2 + 6 5 4 7 > 840 72 10 y 12 4 6 7 6 7 > y > 1 26 7 > 265 1 55 35p 1 2 py 1 + y 4 5 14 y + 3 2 6 y 2 4 y 3y 2 : AppliedMath 2023, 3 390 8 2 3 3 5 2y 2y 395 1979 105p 7 8 > y + 9 y + y 6 7 1 3 15 14 24 4 6 7 6 7 0 y 1 > 2 > 1 + y 4 5 1697 99 63p 35p 9 10 11 > 30 y + y 10 y 18 2 4 11 t (y) = (A60) 2 3 2339 69p 110 1 165 105p 1 5 + 6 7 6 7 > 360 44 7 y 2 4 y 6 7 > y > 1 6 7 1 + y 4 5 > 1697 1 1979 105p 1 395 1 2 py 54 + 21 y + 5 4 3 10 12 2 6 3y 2 y y y Appendix I. Additional Approximations for Arcsine via Iteration The third order iteration, arising from Equation (96), leads to the following approxi- mation for arcsine: sin[s (y)] y sin s (y) y sin[s (y)] y cos[s (y)] n n h (y) = s (y) cos[s (y)] sin[s (y)] y n n cos s y ( ) cos[s (y)] 2 3 sin[s (y)] y sin s (y) y 6 7 sin[s (y)] y cos[s (y)] n n 6 7 (A61) sin s (y) y 4 5 cos[s (y)] sin[s (y)] y n n cos s y ( ) cos[s (y)] 2 3 sin[s (y)] y sin s (y) y 6 7 sin[s (y)] y cos[s (y)] n n 6 7 cos s (y) 4 5 cos[s (y)] sin[s (y)] y n n cos s (y) cos[s (y)] The second order iteration, based on Equation (99), leads to the following approxima- tion for arcsine: h i p p 2 3 4 h y = 1 y y + d y + d y + d y ( ) 2 2,2 2,3 2,4 2 2 h h ii 2 3 4 cos 1 y y + d y + d y + d y y 2,2 2,3 2,4 h h ii 2 2 3 4 sin 1 y y + d y + d y + d y 2,2 2,3 2,4 h i 2 p 3 2 3 4 1 y y + d y + d y + d y + 2,2 2,3 2,4 6 7 h h ii 6 7 6 7 2 2 3 4 cos 1 y y + d y + d y + d y y (A62) cos6 2,2 2,3 2,4 7 y 6 7 h h ii 4 p 5 2 2 3 4 sin 1 y y + d y + d y + d y 2,2 2,3 2,4 h i 2 p 3 2 3 4 1 y y + d y + d y + d y + 2,2 2,3 2,4 6 7 h h ii 6 7 6 2 3 4 7 sin cos 1 y y + d y + d y + d y y 2,2 2,3 2,4 6 7 4 5 h h ii 2 2 3 4 sin 1 y y + d y + d y + d y 2,2 2,3 2,4 Appendix J. Proof of Theorem 7 A zero order spline approximation is simply an afﬁne approximation between the two speciﬁed points. Consistent with the illustration of Figure 13, the zero order spline AppliedMath 2023, 3 391 approximation, denoted f , to asin y , is an afﬁne approximation between the points ( ) (sin(u ), u ) and (sin(v ), v ) leading to 0 0 0 0 v u o o f (y) = u + [y sin(u )] , y 2 [sin(u ), sin(v )]. (A63) 0 o o o o sin(v ) sin(u ) o o With the approximation x = asin(y ) f y it follows, after simpliﬁcation, that o o o o u sin(v ) v sin(u ) + y [v u ] o o o o o o o f y = (A64) ( ) 0 o sin(v ) sin(u ) o o Substitution of u = f (y ) and v = f (y ) yields the required result after the change in 0 L 0 0 U 0 variable from y to y. General Result Consider the general nth order spline approximation f to a function f over the interval [a, b], as given by [16] (eqn. 70): n n n+1 r n+1 r f (x) = (b x) a (x a) + (x a) b (b x) (A65) n n,r n,r å å r=0 r=0 where (ru) 1 f (a) (n + u)! 1 a = , n,r å n+1 (r u)! u!n! (b a) (b a) u=0 (A66) ru (ru) 1 (1) f (b) (n + u)! 1 b = n,r å n+1 (r u)! u!n! (b a) (b a) u=0 The general result stated in Theorem 7 arises with the deﬁnitions f (y) = asin(y), and the interval [a, b] where a = sin(u ), b = sin(v ) and u = f (y ), v = f (y ). The 0 0 0 L 0 0 U 0 approximation is n+1 sin v y [ ( ) ] f (y) = n+1 sin v sin u [ ( ) ( )] o o " # (ru) n r f [sin(u )] (n + u)! 1 r o å [y sin(u )] å + (r u)! u!n! sin v sin u [ ( ) ( )] r=0 u=0 o o (A67) n+1 [y sin(u )] n+1 [sin(v ) sin(u )] o o 2 3 ru (ru) 1) f [sin(v ) n r o (n + u)! 1 4 5 [sin(v ) y] å o å (r u)! u!n! [sin(v ) sin(u )] r=0 u=0 o o (k) for y 2 [sin(v ), sin(v )] and where f is the kth derivative of arcsine. An analytical 0 0 (k) (1) 2 (k) expression for f arises from noting that f (y) = 1/ 1 y and that f has the form " # k+i+1 1 + (1) d[k, i] y k1 (k) f (y) = , k 2 f1, 2, . . .g (A68) k + i + 1 1 i=0 2 2 (1 y ) (k+1) where the coefﬁcients d[k, i] are to be determined. By considering the forms for f (y) (k) and f (y), the algorithm for the coefﬁcients, as speciﬁed in Theorem 7, can be deter- (k) (0) mined. Qi and Zheng [28] detail an alternative form for f . As f [sin(u )] = u and o o (0) f [sin(v )] = v , it then follows that o o AppliedMath 2023, 3 392 n+1 sin v y [ ( ) ] f (y) = n+1 [sin(v ) sin(u )] o o 2 3 (n+r)!u 6 7 n r!n![sin(v ) sin(u )] o o 6 7 [y sin(u )] 6 7+ å o (ru) 4 r1 5 r=0 f [sin(u )] (n + u)! 1 å . . (ru)! u!n! [sin(v ) sin(u )] u=0 o o n+1 (A69) [y sin(u )] n+1 sin v sin u [ ( ) ( )] o o 2 3 (n+r)!v 6 7 r!n![sin(v ) sin(u )] o o 6 7 6 i 7 [sin(v ) y] 6 7 å o ru (ru) 1) f [sin(v ) 6 7 r1 o r=0 (n + u)! 1 4 5 å . . (ru)! u!n! [sin(v ) sin(u )] u=0 o o for y 2 [sin(u ), sin(v )]. The required result follows: the approximation for asin(y ) arises o o o for the case of y = y . Appendix K. Fourth and Sixth Order Approximations for Arccosine Squared The fourth and sixth order approximations for arccosine squared, consistent with Theorem 9, are: p y y c (y) = (1 y) 1 1 (A70) 2,4 Õ 4 r r 4i 4i i=1 16, 732, 749 j6, 808, 161 1, 299, 161 j25, 525, 407 r = + r = + 41 42 12, 500, 000 6, 250, 000 12, 500, 000 12, 500, 000 (A71) 1, 168, 741 j23, 807, 729 16, 131, 473 j9, 610, 843 r = + r = + 43 44 781, 250 12, 500, 000 6, 250, 000 12, 500, 000 2 6 p y y c y = 1 y 1 1 (A72) ( ) ( ) 2,6 4 r r 6i i=1 6i 333, 602, 739 j675, 965, 943 788, 537, 601 j183, 898, 863 r = + r = + 61 62 9 9 250, 000, 000 125, 000, 000 10 10 117, 196, 479 j117, 896, 643 1, 129, 571, 433 j365, 814, 027 r = + r = + (A73) 9 9 10 62, 500, 000 10 200, 000, 000 496, 879, 191 j82, 357, 137 1, 238, 163, 489 j478, 997, 641 r = + r = + 65 66 250, 000, 000 62, 500, 000 500, 000, 000 10 Appendix L. Approximations for Even Powers of Arcsine Borwein [23] (eqn. 2.2 to 2.4) details approximations for even powers of arcsine and the approximations for powers of two, four, six, eight and ten are: n 2 2k 2k 1 2 [k!] y asin(y) S (y) = (A74) 2,n å 2 k (2k)! k=1 " # n k1 2k 2k 3 1 2 [k!] y asin(y) S (y) = (A75) 4,n å å 2 2 m k (2k)! m=1 k=1 AppliedMath 2023, 3 393 " # n k1 m1 2k 2k 45 1 1 2 [k!] y asin(y) S (y) = (A76) 6,n å å å 2 2 2 4 m p k (2k)! k=1 m=1 p=1 " # p1 n k1 m1 2 2k 2k 315 1 1 1 2 [k!] y asin(y) S (y) = (A77) 8,n å å å å 2 2 2 2 2 m p q k (2k)! k=1 m=1 p=1 q=1 " # p1 q1 2 n k1 m1 2k 2k 10! 1 1 1 1 2 [k!] y asin(y) S (y) = . (A78) 10,n å å å å å 5 2 2 2 2 2 4 m p q r k 2k ! ( ) m=1 p=1 q=1 r=1 k=1 Appendix M. Second and Third Order Approximations for Inverse Tangent Integral Second and third order approximations for the inverse tangent integral are: h i p 32 15p T (y) = ln y + 1 + y + + 8p atan(y)+ 2 3 4 2 2 15p y 35 15p y 8 9p + p + + 8p + (A79) 4 3 4 1 + y 1 + y 2 2 3 40 15p 15p y 80 5p y + + 5p + 3/2 3/2 3 2 4 2 9 2 2 (1 + y ) (1 + y ) h i p 788 743p 455p p y T y = ln y + 1 + y + + atan y + p + ( ) ( ) 2 9 12 16 4 1 + y 2 2 979 62p 35p y 472 105p y + + + 115p + 3/2 45 3 4 1 + y 3 2 2 (1 + y ) 3 2 944 230p y 275p 525p y + 35p + 108 + (A80) 3/2 9 3 2 4 16 (1 + y ) (1 + y ) 2 3 2 324 165p 315p y 2744 427p 35p y + + + + 2 5/2 5 4 16 45 12 2 2 (1 + y ) 1 + y ( ) 2 3 2 5 10, 976 427p 70p y 21, 952 854p 28p y + + + 5/2 5/2 2 2 135 9 3 675 45 3 (1 + y ) (1 + y ) References 1. Boyer, C.B. A History of Mathematics; John Wiley: New York, NY, USA, 1991. 2. Bercu, G. The natural approach of trigonometric inequalities—Padé approximant. J. Math. Inequalities 2017, 11, 181–191. [CrossRef] 3. 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AppliedMath – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 4, 2023

**Keywords: **arcsine; arccosine; arctangent; two point spline approximation; upper and lower bounded functions; Newton-Raphson

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