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21: 8 Incomplete Gamma and Related
Functions

Chapter 8 Incomplete Gamma and Related Functions

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22: 19.37 Tables

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Functions $F(\varphi ,k)$ and $E(\varphi ,k)$

►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 10D by Fettis and Caslin (1964). ►Tabulated for $\varphi =0(1^{\circ }){90}^{\circ }$, $k^2=0(.01)1$ to 7S by Beli͡akov et al. (1962). … ►Tabulated for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(1^{\circ }){90}^{\circ }$ to 6D by Byrd and Friedman (1971), for $\varphi =0(5^{\circ }){90}^{\circ }$, $\arcsin k=0(2^{\circ }){90}^{\circ }$ and $5^{\circ }({10}^{\circ }){85}^{\circ }$ to 8D by Abramowitz and Stegun (1964, Chapter 17), and for $\varphi =0({10}^{\circ }){90}^{\circ }$, $\arcsin k=0(5^{\circ }){90}^{\circ }$ to 9D by Zhang and Jin (1996, pp. 674–675). … ►Tabulated for $\varphi =5^{\circ }(5^{\circ }){80}^{\circ }({2.5}^{\circ }){90}^{\circ }$, ${\alpha }^2=-1(.1)-0.1,0.1(.1)1$, $k^2=0(.05)0.9(.02)1$ to 10D by Fettis and Caslin (1964) (and warns of inaccuracies in Selfridge and Maxfield (1958) and Paxton and Rollin (1959)). …

23: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

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24: Software Index

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	Open Source													With Book						Commercial
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7.25(iv) $C\left(x\right)$, $S\left(x\right)$, $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in ℝ$	✓		✓						✓			a	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$	✓											a	✓								✓	✓	✓
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8 Incomplete Gamma and Related Functions
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9 Airy and Related Functions
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34 3j, 6j, 9j Symbols
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25: 12.14 The Function $W(a,x)$

… ►For the modulus functions $\tilde{F}(a,x)$ and $\tilde{G}(a,x)$ see §12.14(x). … ►Other expansions, involving $\cos \left(\frac{1}{4}{x}^2\right)$ and $\sin \left(\frac{1}{4}{x}^2\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-\mathrm{i}a$ and $z$ by $x{\mathrm{e}}^{\pi \mathrm{i}/4}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). … ►uniformly for $t\in [-1+\delta ,\mathrm{\infty })$, with $\zeta$, $\varphi (\zeta )$, $A_s(\zeta )$, and $B_s(\zeta )$ as in §12.10(vii). … ►where $k$ is defined in (12.14.5), and $\tilde{F}(a,x)$ ($>$0), $\tilde{\theta }(a,x)$, $\tilde{G}(a,x)$ ($>$0), and $\tilde{\psi }(a,x)$ are real. $\tilde{F}$ or $\tilde{G}$ is the modulus and $\tilde{\theta }$ or $\tilde{\psi }$ is the corresponding phase. …

26: 21.5 Modular Transformations

… ►Let $𝐀$, $𝐁$, $𝐂$, and $𝐃$ be $g\times g$ matrices with integer elements such that …Here $\xi (𝚪)$ is an eighth root of unity, that is, ${(\xi (𝚪))}^8=1$. … ►($𝐀$ invertible with integer elements.) …($𝐁$ symmetric with integer elements and even diagonal elements.) …($𝐁$ symmetric with integer elements.) …

27: 3.5 Quadrature

… ►If in addition $f$ is periodic, $f\in C^k(ℝ)$, and the integral is taken over a period, then … ►If $f\in C^{2m+2}[a,b]$, then the remainder $E_n(f)$ in (3.5.2) can be expanded in the form … ►About $2^9=512$ function evaluations are needed. … ►For further information, see Mason and Handscomb (2003, Chapter 8), Davis and Rabinowitz (1984, pp. 74–92), and Clenshaw and Curtis (1960). … ►For $C^{\mathrm{\infty }}$ functions Gauss quadrature can be very efficient. …

28: 20.15 Tables

… ►This reference gives ${\theta }_j(x,q)$, $j=1,2,3,4$, and their logarithmic $x$-derivatives to 4D for $x/\pi =0(.1)1$, $\alpha =0(9^{\circ }){90}^{\circ }$, where $\alpha$ is the modular angle given by … ►Spenceley and Spenceley (1947) tabulates ${\theta }_1(x,q)/{\theta }_2(0,q)$, ${\theta }_2(x,q)/{\theta }_2(0,q)$, ${\theta }_3(x,q)/{\theta }_4(0,q)$, ${\theta }_4(x,q)/{\theta }_4(0,q)$ to 12D for $u=0(1^{\circ }){90}^{\circ }$, $\alpha =0(1^{\circ }){89}^{\circ }$, where $u=2x/(\pi {\theta }_3^2(0,q))$ and $\alpha$ is defined by (20.15.1), together with the corresponding values of ${\theta }_2(0,q)$ and ${\theta }_4(0,q)$. ►Lawden (1989, pp. 270–279) tabulates ${\theta }_j(x,q)$, $j=1,2,3,4$, to 5D for $x=0(1^{\circ }){90}^{\circ }$, $q=0.1(.1)0.9$, and also $q$ to 5D for $k^2=0(.01)1$. ►Tables of Neville’s theta functions ${\theta }_s(x,q)$, ${\theta }_c(x,q)$, ${\theta }_d(x,q)$, ${\theta }_n(x,q)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\epsilon ,\alpha =0(5^{\circ }){90}^{\circ }$, where (in radian measure) $\epsilon =x/{\theta }_3^2(0,q)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …

29: 18.5 Explicit Representations

… ►In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_n$. … ►In this equation $w(x)$ is as in Table 18.3.1, (reproduced in Table 18.5.1), and $F(x)$, ${\kappa }_n$ are as in Table 18.5.1. … ►For the definitions of ${}_2{}^{}F_1^{}$, ${}_1{}^{}F_1^{}$, and ${}_2{}^{}F_0^{}$ see §16.2. … ►Similarly in the cases of the ultraspherical polynomials $C_n^{(\lambda )}\left(x\right)$ and the Laguerre polynomials $L_n^{(\alpha )}\left(x\right)$ we assume that $\lambda >-\frac{1}{2},\lambda \ne 0$, and $\alpha >-1$, unless stated otherwise. … ►

$T_4\left(x\right)=8x^4-8x^2+1,$

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30: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form … ►Given $P$, calculate $2P$, $4P$, $8P$ by doubling as above. …If any of $2P$, $4P$, $8P$ is not an integer, then the point has infinite order. Otherwise observe any equalities between $P$, $2P$, $4P$, $8P$, and their negatives. The order of a point (if finite and not already determined) can have only the values 3, 5, 6, 7, 9, 10, or 12, and so can be found from $2P=-P$, $4P=-P$, $4P=-2P$, $8P=P$, $8P=-P$, $8P=-2P$, or $8P=-4P$. …