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11: 23.15 Definitions

§23.15 Definitions

… ►The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … ►If, as a function of $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. If, in addition, $f(\tau )\to 0$ as $q\to 0$, then $f(\tau )$ is called a cusp form. …

12: 28.12 Definitions and Basic Properties

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§28.12(ii) Eigenfunctions ${\mathrm{me}}_{\nu }(z,q)$

… ►However, these functions are not the limiting values of ${\mathrm{me}}_{\pm \nu }(z,q)$ as $\nu \to n$ $(\ne 0)$. … ►Again, the limiting values of ${\mathrm{ce}}_{\nu }(z,q)$ and ${\mathrm{se}}_{\nu }(z,q)$ as $\nu \to n$ $(\ne 0)$ are not the functions ${\mathrm{ce}}_n(z,q)$ and ${\mathrm{se}}_n(z,q)$ defined in §28.2(vi). …

13: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

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§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

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

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14: 26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. … … ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer $k$. … ►

§26.9(ii) Generating Functions

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§26.9(iv) Limiting Form

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15: 14.20 Conical (or Mehler) Functions

… ►(14.2.2) takes the form … … ►

§14.20(v) Trigonometric Expansion

… ►From (14.20.9) or (14.20.10) it is evident that ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(\cos \theta \right)$ is positive for real $\theta$. … ►where the inverse trigonometric functions take their principal values. …

16: 6.16 Mathematical Applications

… ►These limits are not approached uniformly, however. The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi$ and equals $(1.1789\mathrm{\dots })\times \frac{1}{4}\pi$; compare Figure 6.3.2. Hence if $x=\pi /(2n)$ and $n\to \mathrm{\infty }$, then the limiting value of $S_n(x)$ overshoots $\frac{1}{4}\pi$ by approximately 18%. Similarly if $x=\pi /n$, then the limiting value of $S_n(x)$ undershoots $\frac{1}{4}\pi$ by approximately 10%, and so on. … ► …

17: 5.12 Beta Function

§5.12 Beta Function

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Euler’s Beta Integral

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

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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

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5.12.5 $${\int }_0^{\pi /2}{(\cos t)}^{a-1}\cos \left(bt\right)dt=\frac{\pi }{2^a}\frac{1}{a\mathrm{B}(\frac{1}{2}(a+b+1),\frac{1}{2}(a-b+1))},$$ $\mathrm{\Re }a>0$.

ⓘ

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, $\mathrm{\Re }$: real part, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§10.22(ii), §5.12, §7.6(ii)

Permalink:

http://dlmf.nist.gov/5.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.12, §5.12 and Ch.5

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Pochhammer’s Integral

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18: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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§20.2(ii) Periodicity and Quasi-Periodicity

… ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. … ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iv) $z$-Zeros

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19: 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. In particular, ${\psi }^{\prime }\left(z\right)$ is the trigamma function; ${\psi }^{\prime \prime }$, ${\psi }^{(3)}$, ${\psi }^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For $B_{2k}$ see §24.2(i). …

20: 9.12 Scorer Functions

§9.12 Scorer Functions

… ►where … ►

§9.12(ii) Graphs

… ► … ►

Functions and Derivatives

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