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11: 5.1 Special Notation

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$j,m,n$	nonnegative integers.
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$x,y$	real variables.
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$a,b,q,s,w$	real or complex variables with .
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12: 15.14 Integrals

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15.14.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}𝐅(\genfrac{}{}{0pt}{}{a,b}{c};-x)dx=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(a-s\right)\mathrm{\Gamma }\left(b-s\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-s\right)},$$ $\min (\mathrm{\Re }a,\mathrm{\Re }b)>\mathrm{\Re }s>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $s$: nonnegative integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.14

Permalink:

http://dlmf.nist.gov/15.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.14 and Ch.15

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13: 12.13 Sums

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12.13.1 $$U(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{{(-y)}^m}{m!}U(a-m,x),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $U(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.2 $$U(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right)y^{m}U(a+m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.3 $$V(a,x+y)={\mathrm{e}}^{\frac{1}{2}xy+\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{a-\frac{1}{2}}{m}\right)y^{m}V(a-m,x),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\mathrm{e}$: base of natural logarithm, $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Permalink:

http://dlmf.nist.gov/12.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.4 $$V(a,x+y)={\mathrm{e}}^{-\frac{1}{2}xy-\frac{1}{4}{y}^2}\sum _{m=0}^{\mathrm{\infty }}\frac{y^m}{m!}V(a+m,x).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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12.13.5 $$U(a,x\cos t+y\sin t)={\mathrm{e}}^{\frac{1}{4}{(x\sin t-y\cos t)}^2}\sum _{m=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0.0pt}{}{-a-\frac{1}{2}}{m}\right){(\tan t)}^{m}U(m+a,x)U(-m-\frac{1}{2},y),$$ $\mathrm{\Re }a\le -\frac{1}{2},0\le t\le \frac{1}{4}\pi$.

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $\tan z$: tangent function, $x$: real variable, $y$: real variable and $a$: real or complex parameter

Referenced by:

§12.13(i)

Permalink:

http://dlmf.nist.gov/12.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §12.13(i), §12.13 and Ch.12

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14: 4.32 Inequalities

… ►For $x$ real, ►

4.32.1 $$\cosh x\le {\left(\frac{\sinh x}{x}\right)}^3,$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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4.32.2 $x>0$,

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Symbols:

$\cos z$: cosine function, $\tanh z$: hyperbolic tangent function, $\sin z$: sine function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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4.32.3 $$|\cosh x-\cosh y|\ge |x-y|\sqrt{\sinh x\sinh y},$$ $x>0$, $y>0$,

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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4.32.4 $$\arctan x\le \frac{1}{2}\pi \tanh x,$$ $x\ge 0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\tanh z$: hyperbolic tangent function, $\arctan z$: arctangent function and $x$: real variable

Permalink:

http://dlmf.nist.gov/4.32.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.32 and Ch.4

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15: 12.3 Graphics

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§12.3(i) Real Variables

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Figure 12.3.8: $V(a,x)$, $-2.5\le a\le 2.5$, $-2.5\le x\le 2.5$. Magnify 3D Help

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16: 12.1 Special Notation

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$x,y$	real variables.
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$a,\nu$	real or complex parameters.
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17: 32.5 Integral Equations

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32.5.1 $$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

►where $k$ is a real constant, and $\mathrm{Ai}\left(z\right)$ is defined in §9.2. … ►

32.5.2 $$w(z)=K(z,z),$$

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Symbols:

$z$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

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32.5.3 $$w(z)\sim k\mathrm{Ai}\left(z\right),$$ $z\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $z$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

18: 25.14 Lerch’s Transcendent

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25.14.1 $$\mathrm{\Phi }(z,s,a)\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(a+n)}^s},$$ ; $\mathrm{\Re }s>1,|z|=1$.

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Defines:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent

Symbols:

$\equiv$: equals by definition, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.1), p. 27)

Referenced by:

(25.14.2), (25.14.3), §25.14(ii), §25.14, Erratum (V1.0.21) for Equation (25.14.1)

Permalink:

http://dlmf.nist.gov/25.14.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.21):

The previous constraint $a\ne 0,-1,-2,\mathrm{\dots },$ was removed. A clarification regarding the correct constraints for Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$ has been added in the text immediately below.

See also:

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

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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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25.14.3 $${\mathrm{Li}}_s\left(z\right)=z\mathrm{\Phi }(z,s,1),$$ $\mathrm{\Re }s>1$, $|z|\le 1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $\mathrm{\Re }$: real part, $s$: complex variable and $z$: complex variable

Keywords:

specialization

Proof sketch:

Derivable from (25.12.10) and (25.14.1).

Referenced by:

(25.12.12)

Permalink:

http://dlmf.nist.gov/25.14.E3

Encodings:

pMML, png, TeX

See also:

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

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25.14.5 $$\mathrm{\Phi }(z,s,a)=\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}{\mathrm{e}}^{-ax}}{1-z{\mathrm{e}}^{-x}}dx,$$ $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $z=1$; $\mathrm{\Re }s>0$, $\mathrm{\Re }a>0$ if $z\in ℂ\setminus [1,\mathrm{\infty })$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $[a,b)$: half-closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $\setminus$: set subtraction, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.3), p. 27)

Referenced by:

Erratum (V1.1.4) for Equation (25.14.5)

Permalink:

http://dlmf.nist.gov/25.14.E5

Encodings:

pMML, png, TeX

See also:

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

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25.14.6 $$\mathrm{\Phi }(z,s,a)=\frac{1}{2}{a}^{-s}+{\int }_0^{\mathrm{\infty }}\frac{z^x}{{(a+x)}^s}dx-2{\int }_0^{\mathrm{\infty }}\frac{\sin \left(x\ln z-s\arctan \left(x/a\right)\right)}{{(a^2+x^2)}^{s/2}({\mathrm{e}}^{2\pi x}-1)}dx,$$ $\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$.

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Symbols:

$\mathrm{\Phi }(z,s,a)$: Lerch’s transcendent, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\arctan z$: arctangent function, $\ln z$: principal branch of logarithm function, $\mathrm{\Re }$: real part, $\sin z$: sine function, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Erdélyi et al. (1953a, (1.11.4), p. 28)

Notes:

For the case see Erdélyi et al. (1953a, (1.11.4), p. 28). In the case $\left|z\right|=1$ one checks that the first integral converges absolutely iff $\mathrm{\Re }s>1$.

Referenced by:

Erratum (V1.1.4) for Equation (25.14.6)

Permalink:

http://dlmf.nist.gov/25.14.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.4):

The constraint which originally read “$\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $|z|=1,\mathrm{\Re }a>0$” has been improved to be “$\mathrm{\Re }a>0$ if ; $\mathrm{\Re }s>1$, $\mathrm{\Re }a>0$ if $\left|z\right|=1$”.

Suggested 2021-08-23 by Gergő Nemes

See also:

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

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19: 15.1 Special Notation

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$x$	real variable.
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$a,b,c$	real or complex parameters.
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15.1.1 $${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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15.1.2 $$\frac{F(a,b;c;z)}{\mathrm{\Gamma }\left(c\right)}=𝐅(a,b;c;z)=𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={}_2{}^{}𝐅_1^{}(a,b;c;z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

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20: 25.13 Periodic Zeta Function

… ►The notation $F(x,s)$ is used for the polylogarithm ${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}x}\right)$ with $x$ real: ►

25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

►where $\mathrm{\Re }s>1$ if $x$ is an integer, $\mathrm{\Re }s>0$ otherwise. … ►

25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connection formula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25