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1: 23.15 Definitions

§23.15 Definitions

… ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …(Some references refer to $2\mathrm{\ell }$ as the level). …If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. … ►

2: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

►

§14.19(i) Introduction

… ►This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta ,\theta ,\varphi )$, which are related to Cartesian coordinates $(x,y,z)$ by … ►

§14.19(iv) Sums

… ►

§14.19(v) Whipple’s Formula for Toroidal Functions

…

3: 9.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k$	nonnegative integer, except in §9.9(iii).
…
primes	derivatives with respect to argument.

►The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

4: 9.12 Scorer Functions

§9.12 Scorer Functions

… ► $-\mathrm{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ on the interval . … ► … ►

Functions and Derivatives

►As $z\to \mathrm{\infty }$, and with $\delta$ denoting an arbitrary small positive constant, …

5: 5.15 Polygamma Functions

§5.15 Polygamma Functions

►The functions ${\psi }^{(n)}\left(z\right)$, $n=1,2,\mathrm{\dots }$, are called the polygamma functions. …Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►As $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$ …For $B_{2k}$ see §24.2(i). …

6: 20.2 Definitions and Periodic Properties

… ►

§20.2(i) Fourier Series

… ►Corresponding expansions for ${\theta }_j^{\prime }\left(z\right|\tau )$, $j=1,2,3,4$, can be found by differentiating (20.2.1)–(20.2.4) with respect to $z$. ►

§20.2(ii) Periodicity and Quasi-Periodicity

… ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iv) $z$-Zeros

…

7: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

… ►

§15.2(ii) Analytic Properties

… ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►which sometimes needs to be used in §15.4. … ►In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.

8: 5.12 Beta Function

§5.12 Beta Function

… ►

Euler’s Beta Integral

… ►

5.12.2 $${\int }_0^{\pi /2}{\sin }^{2a-1}\theta {\cos }^{2b-1}\theta d\theta =\frac{1}{2}\mathrm{B}(a,b).$$

ⓘ

Symbols:

$\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $a$: real or complex variable and $b$: real or complex variable

A&S Ref:

6.2.1 and 6.2.2

Referenced by:

§10.22(ii)

Permalink:

http://dlmf.nist.gov/5.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

… ►

Pochhammer’s Integral

… ►where the contour starts from an arbitrary point $P$ in the interval $(0,1)$, circles $1$ and then $0$ in the positive sense, circles $1$ and then $0$ in the negative sense, and returns to $P$. …

9: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►

16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

… ►

16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

ⓘ

Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

… ► …

10: 11.9 Lommel Functions

§11.9 Lommel Functions

… ► … ►

§11.9(ii) Expansions in Series of Bessel Functions

… ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … ►