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1: 20 Theta Functions

Chapter 20 Theta Functions

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2: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …

3: 16.8 Differential Equations

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§16.8(i) Classification of Singularities

… ►Compare §2.7(i) in the case $n=2$. … ►In each case there are no other singularities. … ►Analytical continuation formulas for ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. … ►Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. …

4: 8 Incomplete Gamma and Related
Functions

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5: 28 Mathieu Functions and Hill’s Equation

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6: 17.4 Basic Hypergeometric Functions

… ►In practice this discrepancy does not usually cause serious problems because the case most often considered is $r=s$. … ►The infinite series converge when $s\ge r$ provided that and also, in the case $s=r$, . … ►

§17.4(iv) Classification

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7: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

8: 23 Weierstrass Elliptic and Modular
Functions

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9: Software Index

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	Open Source													With Book						Commercial
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20 Theta Functions
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A Classification of Software

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10: 36 Integrals with Coalescing Saddles

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