Euler constant

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31: 18.25 Wilson Class: Definitions

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$$\gamma ,\delta >-1,\beta >N+\gamma .$$

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►The first four sets imply $\gamma +\delta >-2$, and the last four imply . …

32: 21.5 Modular Transformations

… ►Here $\xi (𝚪)$ is an eighth root of unity, that is, ${(\xi (𝚪))}^8=1$. For general $𝚪$, it is difficult to decide which root needs to be used. The choice depends on $𝚪$, but is independent of $𝐳$ and $𝛀$. … ►where $\kappa (𝜶,𝜷,𝚪)$ is a complex number that depends on $𝜶$, $𝜷$, and $𝚪$. However, $\kappa (𝜶,𝜷,𝚪)$ is independent of $𝐳$ and $𝛀$. …

33: 36.6 Scaling Relations

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$\text{cuspoids: }𝐲(k)=(x_1{k}^{{\gamma }_{1K}},x_2{k}^{{\gamma }_{2K}},\mathrm{\dots },x_K{k}^{{\gamma }_{KK}}),$

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$\text{umbilics: }{\gamma }_x^{(\mathrm{U})}=\frac{2}{3},$

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${\gamma }_y^{(\mathrm{U})}=\frac{2}{3},$

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${\gamma }_z^{(\mathrm{U})}=\frac{1}{3}.$

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Table 36.6.1: Special cases of scaling exponents for cuspoids.

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singularity	$K$	${\beta }_K$	${\gamma }_{1K}$	${\gamma }_{2K}$	${\gamma }_{3K}$	${\gamma }_K$
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34: 15.10 Hypergeometric Differential Equation

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15.10.21 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-a-b\right)}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a+b-c\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}{w}_4(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_4(z)$: Kummer’s solution $w_4$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.25 $$w_1(z)=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(b-a\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}{w}_5(z)+\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-b\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}{w}_6(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_5(z)$: Kummer’s solution $w_5$ and $w_6(z)$: Kummer’s solution $w_6$

Referenced by:

§15.8(i)

Permalink:

http://dlmf.nist.gov/15.10.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.29 $$w_1(z)={\mathrm{e}}^{b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-c+1\right)}{\mathrm{\Gamma }\left(a+b-c+1\right)\mathrm{\Gamma }\left(c-b\right)}{w}_3(z)+{\mathrm{e}}^{(b-c)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(a-c+1\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(a-b+1\right)}{w}_5(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_3(z)$: Kummer’s solution $w_3$ and $w_5(z)$: Kummer’s solution $w_5$

Permalink:

http://dlmf.nist.gov/15.10.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.33 $$w_1(z)={\mathrm{e}}^{(c-a)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(1-b\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{w}_4(z)+{\mathrm{e}}^{-a\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(1-b\right)}{\mathrm{\Gamma }\left(a-b+1\right)\mathrm{\Gamma }\left(c-a\right)}{w}_5(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_4(z)$: Kummer’s solution $w_4$ and $w_5(z)$: Kummer’s solution $w_5$

Permalink:

http://dlmf.nist.gov/15.10.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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15.10.34 $$w_1(z)={\mathrm{e}}^{(c-b)\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(1-a\right)}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a-b+1\right)}{w}_4(z)+{\mathrm{e}}^{-b\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(1-a\right)}{\mathrm{\Gamma }\left(b-a+1\right)\mathrm{\Gamma }\left(c-b\right)}{w}_6(z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $w_1(z)$: Kummer’s solution $w_1$ , $w_4(z)$: Kummer’s solution $w_4$ and $w_6(z)$: Kummer’s solution $w_6$

Permalink:

http://dlmf.nist.gov/15.10.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §15.10(ii), §15.10 and Ch.15

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35: 5.4 Special Values and Extrema

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$\mathrm{\Gamma }\left(1\right)=1,$

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$n!=\mathrm{\Gamma }\left(n+1\right).$

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5.4.11 $${\mathrm{\Gamma }}^{\prime }\left(1\right)=-\gamma .$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $\gamma$: Euler’s constant

A&S Ref:

6.3.2

Referenced by:

§5.4(i)

Permalink:

http://dlmf.nist.gov/5.4.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(i), §5.4 and Ch.5

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$\psi \left(1\right)=-\gamma ,$

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Table 5.4.1: ${\mathrm{\Gamma }}^{\prime }\left(x_n\right)=\psi \left(x_n\right)=0$.

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$n$	$x_n$	$\mathrm{\Gamma }\left(x_n\right)$
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36: 5.18 $q$-Gamma and $q$-Beta Functions

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5.18.5 $${\mathrm{\Gamma }}_q\left(1\right)={\mathrm{\Gamma }}_q\left(2\right)=1,$$

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function and $q$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

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5.18.7 $${\mathrm{\Gamma }}_q\left(z+1\right)=\frac{1-q^z}{1-q}{\mathrm{\Gamma }}_q\left(z\right).$$

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.18.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

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5.18.8

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $x$: real variable

Permalink:

http://dlmf.nist.gov/5.18.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

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5.18.9 $${\mathrm{\Gamma }}_q\left(x\right)>{\mathrm{\Gamma }}_r\left(x\right),$$

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Symbols:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, $q$: real or complex variable and $x$: real variable

Permalink:

http://dlmf.nist.gov/5.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

… ►For the $q$-digamma or $q$-psi function ${\psi }_q(z)={\mathrm{\Gamma }}_q^{\prime }\left(z\right)/{\mathrm{\Gamma }}_q\left(z\right)$ see Salem (2013). …

37: 30.3 Eigenvalues

… ►The eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ are analytic functions of the real variable ${\gamma }^2$ and satisfy ►

30.3.1

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Symbols:

${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.3(ii), §30.3 and Ch.30

… ►where ${\alpha }_k$, ${\beta }_k$, ${\gamma }_k$ are defined by … ►

${\gamma }_k={\gamma }^2,$

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$${\gamma }_k={\gamma }^2\frac{(k-1)k}{(2k+2m-3)(2k+2m-1)}.$$

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38: 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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39: 30.18 Software

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SWF1: ${\lambda }_n^m\left({\gamma }^2\right)$.

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SWF2: ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$.

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SWF3: ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$.

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SWF5: $K_n^m(\gamma )$ in §30.11(v).

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§30.18(ii) Eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$

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40: 30.7 Graphics

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Figure 30.7.1: Eigenvalues ${\lambda }_n^0\left({\gamma }^2\right)$, $n=0,1,2,3$, $-10\le {\gamma }^2\le 10$. Magnify

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Figure 30.7.2: Eigenvalues ${\lambda }_n^1\left({\gamma }^2\right)$ $n=1,2,3,4$, $-10\le {\gamma }^2\le 10$. Magnify

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Figure 30.7.9: ${\mathrm{𝖯𝗌}}_2^0(x,{\gamma }^2)$, $-1\le x\le 1$, $-50\le {\gamma }^2\le 50$. Magnify 3D Help

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Figure 30.7.10: ${\mathrm{𝖯𝗌}}_3^1(x,{\gamma }^2)$, $-1\le x\le 1$, $-50\le {\gamma }^2\le 50$. Magnify 3D Help

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Figure 30.7.15: . Magnify 3D Help

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