relations to other functions

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31—40 of 162 matching pages

31: 8.22 Mathematical Applications

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

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32: 18.11 Relations to Other Functions

§18.11 Relations to Other Functions

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Ultraspherical

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Hermite

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33: 19.25 Relations to Other Functions

§19.25 Relations to Other Functions

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§19.25(iv) Theta Functions

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§19.25(v) Jacobian Elliptic Functions

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§19.25(vi) Weierstrass Elliptic Functions

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34: 33.16 Connection Formulas

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§33.16(i) $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ in Terms of $f$ and $h$

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§33.16(ii) $f$ and $h$ in Terms of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ when $ϵ>0$

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§33.16(iii) $f$ and $h$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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§33.16(iv) $s$ and $c$ in Terms of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ when $ϵ>0$

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§33.16(v) $s$ and $c$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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35: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality … ► … ► …

36: 18.5 Explicit Representations

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Chebyshev

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§18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

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Hermite

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37: 14.7 Integer Degree and Order

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§14.7(i) $\mu =0$

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38: 25.14 Lerch’s Transcendent

… ►The Hurwitz zeta function $\zeta (s,a)$ (§25.11) and the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (§25.12(ii)) are special cases: ►

25.14.2 $$\zeta (s,a)=\mathrm{\Phi }(1,s,a),$$ $\mathrm{\Re }s>1$, $a\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\mathrm{\Re }$: real part, $a$: real or complex parameter and $s$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.11.1), (25.14.1).

Permalink:

http://dlmf.nist.gov/25.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.14(i), §25.14 and Ch.25

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39: 25.12 Polylogarithms

… ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include …and … ►In terms of polylogarithms …

40: 22.2 Definitions

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22.2.9 $$\mathrm{sc}(z,k)=\frac{{\theta }_3(0,q)}{{\theta }_4(0,q)}\frac{{\theta }_1(\zeta ,q)}{{\theta }_2(\zeta ,q)}=\frac{1}{\mathrm{cs}(z,k)}.$$

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Defines:

$\mathrm{cs}(z,k)$: Jacobian elliptic function and $\mathrm{sc}(z,k)$: Jacobian elliptic function

Symbols:

${\theta }_j(z,q)$: theta function, $q$: nome, $z$: complex, $k$: modulus and $\zeta$: change of variable

Referenced by:

§23.6(ii)

Permalink:

http://dlmf.nist.gov/22.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §22.2 and Ch.22

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