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11: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … …

12: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

13: 27.21 Tables

… ►Table III lists all solutions $n\le {10}^4$ of the equation $d\left(n\right)=m$, and Table IV lists all solutions $n$ of the equation $\sigma (n)=m$ for all $m\le {10}^4$. …

14: 25.15 Dirichlet $L$-functions

… ►

25.15.1 $$L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s},$$ $\mathrm{\Re }s>1$,

ⓘ

Defines:

$L(s,\chi )$: Dirichlet $L$-function

Symbols:

$\mathrm{\Re }$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, p. 249)

Referenced by:

(25.11.36), (25.11.36), §25.11(x), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►

25.15.3 $$L(s,\chi )=k^{-s}\sum _{r=1}^{k-1}\chi (r)\zeta (s,\frac{r}{k}),$$

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $\zeta (s,a)$: Hurwitz zeta function, $k$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

finite sum

Source:

Apostol (1976, (8), p. 255)

Notes:

In the original source the upper-index of the finite sum is $k$. We use $k-1$ since $\chi (k)=0$.

Referenced by:

(25.11.36), (25.11.36), §25.11(x), §25.15(i), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.15(i), §25.15 and Ch.25

… ►When $\chi$ is a primitive character (mod $k$) the $L$-functions satisfy the functional equation: … ►

25.15.10

ⓘ

Symbols:

$L(s,\chi )$: Dirichlet $L$-function, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

zeros

Source:

Apostol (1976, Theorem 12.20, p. 268)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.10)

Permalink:

http://dlmf.nist.gov/25.15.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(ii), §25.15 and Ch.25

15: 14.31 Other Applications

… ►

§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►

§14.31(ii) Conical Functions

… ►

§14.31(iii) Miscellaneous

►Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …

16: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. …

17: 27.14 Unrestricted Partitions

… ►

27.14.2 $$f\left(x\right)=\prod _{m=1}^{\mathrm{\infty }}(1-x^m)={(x;x)}_{\mathrm{\infty }},$$ ,

ⓘ

Defines:

$f\left(x\right)$: Euler’s reciprocal function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $m$: positive integer and $x$: real number

Referenced by:

§27.14(iv), §27.21, Erratum (V1.0.28) for Equation (27.14.2)

Permalink:

http://dlmf.nist.gov/27.14.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.28):

The representation in terms of ${(x;x)}_{\mathrm{\infty }}$ was added to this equation.

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

… ►

§27.14(iv) Relation to Modular Functions

… ► $\eta \left(\tau \right)$ satisfies the following functional equation: if $a,b,c,d$ are integers with $ad-bc=1$ and $c>0$, then … ►

§27.14(vi) Ramanujan’s Tau Function

… ►and satisfies the functional equation …

18: 10.13 Other Differential Equations

§10.13 Other Differential Equations

… ►

10.13.4 $$w^{\prime \prime }+\frac{1\mp 2\nu }{z}{w}^{\prime }+{\lambda }^{2}w=0,$$ $w=z^{\pm \nu }{𝒞}_{\nu }\left(\lambda z\right)$,

ⓘ

Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.52

Referenced by:

Erratum (V1.0.8) for Equation (10.13.4)

Permalink:

http://dlmf.nist.gov/10.13.E4

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.8):

This equation has been generalized to allow an additional case. Originally only the case $w=z^{\nu }{𝒞}_{\nu }\left(\lambda z\right)$ was covered.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §10.13 and Ch.10

…

19: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. … ►Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). … ►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

20: 4.7 Derivatives and Differential Equations

… ►For a nonvanishing analytic function $f(z)$, the general solution of the differential equation … ►The general solution of the differential equation …