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31: 25.2 Definition and Expansions

… ►When $\mathrm{\Re }s>1$, … ►

25.2.2 $$\zeta \left(s\right)=\frac{1}{1-2^{-s}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(2n+1)}^s},$$ $\mathrm{\Re }s>1$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.2.1).

A&S Ref:

23.2.20 (is the special case with integer values of $s$)

Referenced by:

(25.11.11)

Permalink:

http://dlmf.nist.gov/25.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

… ►

25.2.6 $${\zeta }^{\prime }\left(s\right)=-\sum _{n=2}^{\mathrm{\infty }}(\ln n)n^{-s},$$ $\mathrm{\Re }s>1$.

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Apostol (1976, (12), p. 236)

Permalink:

http://dlmf.nist.gov/25.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

… ►

25.2.11 $$\zeta \left(s\right)=\prod _p{(1-p^{-s})}^{-1},$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $p$: prime number and $s$: complex variable

Keywords:

infinite product, primes

Source:

Apostol (1976, p. 231)

A&S Ref:

23.2.2

Referenced by:

§25.10(i)

Permalink:

http://dlmf.nist.gov/25.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

… ►product over zeros $\rho$ of $\zeta$ with $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).

32: 13.23 Integrals

… ►

13.23.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)}{{\left(z+\frac{1}{2}\right)}^{\mu +\nu +\frac{1}{2}}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}+\mu +\nu }{1+2\mu };\frac{1}{z+\frac{1}{2}}),$$ $\mathrm{\Re }\mu +\nu +\frac{1}{2}>0$, $\mathrm{\Re }z>\frac{1}{2}$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

►

13.23.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(2\mu +1\right){\left(z+\frac{1}{2}\right)}^{-\kappa -\mu -\frac{1}{2}}{\left(z-\frac{1}{2}\right)}^{\kappa -\mu -\frac{1}{2}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }z>\frac{1}{2}$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

►

13.23.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\kappa -\nu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\nu \right)},$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

… ►

13.23.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{\frac{1}{2}t}{t}^{\nu -1}{W}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +\nu \right)\mathrm{\Gamma }\left(-\kappa -\nu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)},$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

… ►Then for $\mu$ in the half-plane $\mathrm{\Re }\mu \ge {\mu }_1>\max (-{\rho }_0,\mathrm{\Re }\kappa -\frac{1}{2})$ …

33: 4.6 Power Series

… ►

4.6.2 $$\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}{\left(\frac{z-1}{z}\right)}^2+\frac{1}{3}{\left(\frac{z-1}{z}\right)}^3+\mathrm{\cdots },$$ $\mathrm{\Re }z\ge \frac{1}{2}$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.25

Permalink:

http://dlmf.nist.gov/4.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►

4.6.4 $$\ln z=2\left(\left(\frac{z-1}{z+1}\right)+\frac{1}{3}{\left(\frac{z-1}{z+1}\right)}^3+\frac{1}{5}{\left(\frac{z-1}{z+1}\right)}^5+\mathrm{\cdots }\right),$$ $\mathrm{\Re }z\ge 0$, $z\ne 0$,

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

4.1.27

Permalink:

http://dlmf.nist.gov/4.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

… ►

4.6.6 $$\ln \left(z+a\right)=\ln a+2\left(\left(\frac{z}{2a+z}\right)+\frac{1}{3}{\left(\frac{z}{2a+z}\right)}^3+\frac{1}{5}{\left(\frac{z}{2a+z}\right)}^5+\mathrm{\cdots }\right),$$ $a>0$, $\mathrm{\Re }z\ge -a$, $z\ne -a$.

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.1.29

Permalink:

http://dlmf.nist.gov/4.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.6(i), §4.6 and Ch.4

…

34: 6.14 Integrals

… ►

6.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}{E}_1\left(t\right)dt=\frac{1}{a}\ln \left(1+a\right),$$ $\mathrm{\Re }a>-1$,

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.1.34 (special case of)

Permalink:

http://dlmf.nist.gov/6.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

►

6.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{Ci}\left(t\right)dt=-\frac{1}{2a}\ln \left(1+a^2\right),$$ $\mathrm{\Re }a>0$,

ⓘ

Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function and $\mathrm{\Re }$: real part

A&S Ref:

5.2.28

Referenced by:

§6.14(i)

Permalink:

http://dlmf.nist.gov/6.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

►

6.14.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-at}\mathrm{si}\left(t\right)dt=-\frac{1}{a}\arctan a,$$ $\mathrm{\Re }a>0$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\mathrm{\Re }$: real part and $\mathrm{si}\left(z\right)$: sine integral

A&S Ref:

5.2.29

Permalink:

http://dlmf.nist.gov/6.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §6.14(i), §6.14 and Ch.6

…

35: 12.12 Integrals

… ►

12.12.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{\mu -1}U(a,t)dt=\frac{\sqrt{\pi }{2}^{-\frac{1}{2}(\mu +a+\frac{1}{2})}\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}(\mu +a+\frac{3}{2})\right)},$$ $\mathrm{\Re }\mu >0$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

►

12.12.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{3}{4}{t}^2}{t}^{-a-\frac{3}{2}}U(a,t)dt=2^{\frac{1}{4}+\frac{1}{2}a}\mathrm{\Gamma }\left(-a-\frac{1}{2}\right)\cos \left((\frac{1}{4}a+\frac{1}{8})\pi \right),$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

►

12.12.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{4}{t}^2}{t}^{-a-\frac{1}{2}}{(x^2+t^2)}^{-1}U(a,t)dt=\sqrt{\pi /2}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)x^{-a-\frac{3}{2}}{\mathrm{e}}^{\frac{1}{4}{x}^2}U(-a,x),$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $x$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12 and Ch.12

… ►

12.12.4 $${(U(a,z))}^2+{(\overline{U}(a,z))}^2=\frac{2^{\frac{3}{2}}}{\pi }\mathrm{\Gamma }\left(\frac{1}{2}-a\right){\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2at+\frac{1}{2}{z}^2\tanh t}}{\sqrt{\sinh \left(2t\right)}}dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\overline{U}(a,x)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: real or complex parameter

Notes:

See Durand (1975).

Referenced by:

§12.12

Permalink:

http://dlmf.nist.gov/12.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.12, §12.12 and Ch.12

…

36: 10.22 Integrals

… ►When $\mathrm{\Re }\mu >-1$ … ►When $\mathrm{\Re }\mu >0$, … ►When $\mathrm{\Re }\nu >\mathrm{\Re }\mu >-1$, … ►When $\mathrm{\Re }\mu >0$, … ►If $\mathrm{\Re }\nu >-\frac{1}{2}$, then …

37: 13.16 Integral Representations

… ►

13.16.1 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\mu +\frac{1}{2}}{2}^{-2\mu }}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_{-1}^1{\mathrm{e}}^{\frac{1}{2}zt}{(1+t)}^{\mu -\frac{1}{2}-\kappa }{(1-t)}^{\mu -\frac{1}{2}+\kappa }dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\left|\mathrm{\Re }\kappa \right|$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.29(iii)

Permalink:

http://dlmf.nist.gov/13.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

►

13.16.2 $$M_{\kappa ,\mu }\left(z\right)=\frac{\mathrm{\Gamma }\left(1+2\mu \right)z^{\lambda }}{\mathrm{\Gamma }\left(1+2\mu -2\lambda \right)\mathrm{\Gamma }\left(2\lambda \right)}{\int }_0^1{M}_{\kappa -\lambda ,\mu -\lambda }\left(zt\right){\mathrm{e}}^{\frac{1}{2}z(t-1)}{t}^{\mu -\lambda -\frac{1}{2}}{(1-t)}^{2\lambda -1}dt,$$ $\mathrm{\Re }\mu +\frac{1}{2}>\mathrm{\Re }\lambda >0$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§13.23(v)

Permalink:

http://dlmf.nist.gov/13.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

►

13.16.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{\kappa -\frac{1}{2}}{J}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ $\mathrm{\Re }\left(\kappa +\mu \right)+\frac{1}{2}>0$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(i), §13.16 and Ch.13

►

13.16.4 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{-\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\kappa -\frac{1}{2}}{I}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

Erratum (V1.0.5) for Equation (13.16.4)

Permalink:

http://dlmf.nist.gov/13.16.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the condition for the validity of this equation was stated incorrectly as $\mathrm{\Re }(\kappa -\mu )-\frac{1}{2}>0$. The correct condition is .

See also:

Annotations for §13.16(i), §13.16 and Ch.13

… ►where $c$ is arbitrary, $\mathrm{\Re }c>0$. …

38: 5 Gamma Function

…

39: 9 Airy and Related Functions

…

40: 11 Struve and Related Functions

…