# Euler constant

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##### 1: 16.15 Integral Representations and Integrals
16.15.1 ${F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\frac{\Gamma\left(% \gamma\right)}{\Gamma\left(\alpha\right)\Gamma\left(\gamma-\alpha\right)}\int_% {0}^{1}\frac{u^{\alpha-1}(1-u)^{\gamma-\alpha-1}}{(1-ux)^{\beta}(1-uy)^{\beta^% {\prime}}}\mathrm{d}u,$ $\Re\alpha>0$, $\Re\left(\gamma-\alpha\right)>0$,
16.15.2 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=% \frac{\Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left% (\beta\right)\Gamma\left(\beta^{\prime}\right)\Gamma\left(\gamma-\beta\right)% \Gamma\left(\gamma^{\prime}-\beta^{\prime}\right)}\int_{0}^{1}\!\!\!\int_{0}^{% 1}\frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u)^{\gamma-\beta-1}(1-v)^{\gamma^{% \prime}-\beta^{\prime}-1}}{(1-ux-vy)^{\alpha}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\beta>0$, $\Re\gamma^{\prime}>\Re\beta^{\prime}>0$,
16.15.3 ${F_{3}}\left(\alpha,\alpha^{\prime};\beta,\beta^{\prime};\gamma;x,y\right)=% \frac{\Gamma\left(\gamma\right)}{\Gamma\left(\beta\right)\Gamma\left(\beta^{% \prime}\right)\Gamma\left(\gamma-\beta-\beta^{\prime}\right)}\iint_{\Delta}% \frac{u^{\beta-1}v^{\beta^{\prime}-1}(1-u-v)^{\gamma-\beta-\beta^{\prime}-1}}{% (1-ux)^{\alpha}(1-vy)^{\alpha^{\prime}}}\mathrm{d}u\mathrm{d}v,$ $\Re\left(\gamma-\beta-\beta^{\prime}\right)>0$, $\Re\beta>0$, $\Re\beta^{\prime}>0$,
16.15.4 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\frac{% \Gamma\left(\gamma\right)\Gamma\left(\gamma^{\prime}\right)}{\Gamma\left(% \alpha\right)\Gamma\left(\beta\right)\Gamma\left(\gamma-\alpha\right)\Gamma% \left(\gamma^{\prime}-\beta\right)}\int_{0}^{1}\!\!\!\int_{0}^{1}\frac{u^{% \alpha-1}v^{\beta-1}(1-u)^{\gamma-\alpha-1}(1-v)^{\gamma^{\prime}-\beta-1}}{(1% -ux)^{\gamma+\gamma^{\prime}-\alpha-1}(1-vy)^{\gamma+\gamma^{\prime}-\beta-1}(% 1-ux-vy)^{\alpha+\beta-\gamma-\gamma^{\prime}+1}}\mathrm{d}u\mathrm{d}v,$ $\Re\gamma>\Re\alpha>0$, $\Re\gamma^{\prime}>\Re\beta>0$.
##### 2: 30.1 Special Notation
 $x$ real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, $-1. real parameter (positive, zero, or negative). …
Flammer (1957) and Abramowitz and Stegun (1964) use $\lambda_{mn}(\gamma)$ for $\lambda^{m}_{n}\left(\gamma^{2}\right)+\gamma^{2}$, $R_{mn}^{(j)}(\gamma,z)$ for $S^{m(j)}_{n}\left(z,\gamma\right)$, and
$S^{(1)}_{mn}(\gamma,x)=d_{mn}(\gamma)\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}% \right),$
$S^{(2)}_{mn}(\gamma,x)=d_{mn}(\gamma)\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}% \right),$
where $d_{mn}(\gamma)$ is a normalization constant determined by …
##### 3: 30.5 Functions of the Second Kind
Other solutions of (30.2.1) with $\mu=m$, $\lambda=\lambda^{m}_{n}\left(\gamma^{2}\right)$, and $z=x$ are
30.5.1 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right),$ $n=m,m+1,m+2,\dots$.
30.5.2 $\mathsf{Qs}^{m}_{n}\left(-x,\gamma^{2}\right)=(-1)^{n-m+1}\mathsf{Qs}^{m}_{n}% \left(x,\gamma^{2}\right),$
30.5.4 $\mathscr{W}\left\{\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right),\mathsf{Qs}^{m}% _{n}\left(x,\gamma^{2}\right)\right\}=\frac{(n+m)!}{(1-x^{2})(n-m)!}A_{n}^{m}(% \gamma^{2})A_{n}^{-m}(\gamma^{2})\quad(\neq 0),$
with $A_{n}^{\pm m}(\gamma^{2})$ as in (30.11.4). …
##### 4: 5.22 Tables
Abramowitz and Stegun (1964, Chapter 6) tabulates $\Gamma\left(x\right)$, $\ln\Gamma\left(x\right)$, $\psi\left(x\right)$, and $\psi'\left(x\right)$ for $x=1(.005)2$ to 10D; $\psi''\left(x\right)$ and $\psi^{(3)}\left(x\right)$ for $x=1(.01)2$ to 10D; $\Gamma\left(n\right)$, $\ifrac{1}{\Gamma\left(n\right)}$, $\Gamma\left(n+\tfrac{1}{2}\right)$, $\psi\left(n\right)$, $\operatorname{log}_{10}\Gamma\left(n\right)$, $\operatorname{log}_{10}\Gamma\left(n+\tfrac{1}{3}\right)$, $\operatorname{log}_{10}\Gamma\left(n+\tfrac{1}{2}\right)$, and $\operatorname{log}_{10}\Gamma\left(n+\tfrac{2}{3}\right)$ for $n=1(1)101$ to 8–11S; $\Gamma\left(n+1\right)$ for $n=100(100)1000$ to 20S. Zhang and Jin (1996, pp. 67–69 and 72) tabulates $\Gamma\left(x\right)$, $\ifrac{1}{\Gamma\left(x\right)}$, $\Gamma\left(-x\right)$, $\ln\Gamma\left(x\right)$, $\psi\left(x\right)$, $\psi\left(-x\right)$, $\psi'\left(x\right)$, and $\psi'\left(-x\right)$ for $x=0(.1)5$ to 8D or 8S; $\Gamma\left(n+1\right)$ for $n=0(1)100(10)250(50)500(100)3000$ to 51S. … Abramov (1960) tabulates $\ln\Gamma\left(x+iy\right)$ for $x=1$ ($.01$) $2$, $y=0$ ($.01$) $4$ to 6D. Abramowitz and Stegun (1964, Chapter 6) tabulates $\ln\Gamma\left(x+iy\right)$ for $x=1$ ($.1$) $2$, $y=0$ ($.1$) $10$ to 12D. …Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of $\Gamma\left(x+iy\right)$, $\ln\Gamma\left(x+iy\right)$, and $\psi\left(x+iy\right)$ for $x=0.5,1,5,10$, $y=0(.5)10$ to 8S.
##### 5: 30.6 Functions of Complex Argument
The solutions
$\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right),$
$\mathit{Qs}^{m}_{n}\left(z,\gamma^{2}\right),$
30.6.3 $\mathscr{W}\left\{\mathit{Ps}^{m}_{n}\left(z,\gamma^{2}\right),\mathit{Qs}^{m}% _{n}\left(z,\gamma^{2}\right)\right\}=\frac{(-1)^{m}(n+m)!}{(1-z^{2})(n-m)!}A_% {n}^{m}(\gamma^{2})A_{n}^{-m}(\gamma^{2}),$
with $A_{n}^{\pm m}(\gamma^{2})$ as in (30.11.4). …
##### 6: 16.16 Transformations of Variables
16.16.3 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\alpha;x,y\right)=(1-y)^{-% \beta^{\prime}}{F_{1}}\left(\beta;\alpha-\beta^{\prime},\beta^{\prime};\gamma;% x,\frac{x}{1-y}\right),$
16.16.4 ${F_{3}}\left(\alpha,\gamma-\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=(1-y)% ^{-\beta^{\prime}}{F_{1}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,\frac{y}{y% -1}\right),$
16.16.7 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x(1-y),y(1-x)\right)=\sum_{k=% 0}^{\infty}\frac{{\left(\alpha\right)_{k}}{\left(\beta\right)_{k}}{\left(% \alpha+\beta-\gamma-\gamma^{\prime}+1\right)_{k}}}{{\left(\gamma\right)_{k}}{% \left(\gamma^{\prime}\right)_{k}}k!}x^{k}y^{k}{{}_{2}F_{1}}\left({\alpha+k,% \beta+k\atop\gamma+k};x\right){{}_{2}F_{1}}\left({\alpha+k,\beta+k\atop\gamma^% {\prime}+k};y\right);$
16.16.9 ${F_{2}}\left(\alpha;\beta,\beta^{\prime};\gamma,\gamma^{\prime};x,y\right)=(1-% x)^{-\alpha}{F_{2}}\left(\alpha;\gamma-\beta,\beta^{\prime};\gamma,\gamma^{% \prime};\frac{x}{x-1},\frac{y}{1-x}\right),$
16.16.10 ${F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\frac{\Gamma\left(% \gamma^{\prime}\right)\Gamma\left(\beta-\alpha\right)}{\Gamma\left(\gamma^{% \prime}-\alpha\right)\Gamma\left(\beta\right)}(-y)^{-\alpha}{F_{4}}\left(% \alpha,\alpha-\gamma^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\Gamma\left(\gamma^{\prime}\right)\Gamma\left(\alpha-\beta\right% )}{\Gamma\left(\gamma^{\prime}-\beta\right)\Gamma\left(\alpha\right)}(-y)^{-% \beta}{F_{4}}\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-\alpha+1;\frac{x% }{y},\frac{1}{y}\right).$
##### 7: 8.8 Recurrence Relations and Derivatives
If $w(a,z)=\gamma\left(a,z\right)$ or $\Gamma\left(a,z\right)$, then …
8.8.8 $\gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}{\Gamma\left(a-n\right)}% \gamma\left(a-n,z\right)-z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a% \right)}{\Gamma\left(a-k\right)}z^{-k},$
8.8.10 $\Gamma\left(a,z\right)=\frac{\Gamma\left(a\right)}{\Gamma\left(a-n\right)}% \Gamma\left(a-n,z\right)+z^{a-1}e^{-z}\sum_{k=0}^{n-1}\frac{\Gamma\left(a% \right)}{\Gamma\left(a-k\right)}z^{-k},$
##### 8: 8.2 Definitions and Basic Properties
The general values of the incomplete gamma functions $\gamma\left(a,z\right)$ and $\Gamma\left(a,z\right)$ are defined by … In this subsection the functions $\gamma$ and $\Gamma$ have their general values. The function $\gamma^{*}\left(a,z\right)$ is entire in $z$ and $a$. … If $w=\gamma\left(a,z\right)$ or $\Gamma\left(a,z\right)$, then …
##### 9: 5.8 Infinite Products
5.8.1 $\Gamma\left(z\right)=\lim_{k\to\infty}\frac{k!k^{z}}{z(z+1)\cdots(z+k)},$ $z\neq 0,-1,-2,\dots$,
5.8.3 $\left|\frac{\Gamma\left(x\right)}{\Gamma\left(x+\mathrm{i}y\right)}\right|^{2}% =\prod_{k=0}^{\infty}\left(1+\frac{y^{2}}{(x+k)^{2}}\right),$ $x\neq 0,-1,\dots$.
5.8.5 $\prod_{k=0}^{\infty}\frac{(a_{1}+k)(a_{2}+k)\cdots(a_{m}+k)}{(b_{1}+k)(b_{2}+k% )\cdots(b_{m}+k)}=\frac{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\cdots% \Gamma\left(b_{m}\right)}{\Gamma\left(a_{1}\right)\Gamma\left(a_{2}\right)% \cdots\Gamma\left(a_{m}\right)},$
##### 10: 8.7 Series Expansions
8.7.1 $\gamma^{*}\left(a,z\right)=e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma\left(a% +k+1\right)}=\frac{1}{\Gamma\left(a\right)}\sum_{k=0}^{\infty}\frac{(-z)^{k}}{% k!(a+k)}.$
8.7.2 $\gamma\left(a,x+y\right)-\gamma\left(a,x\right)=\Gamma\left(a,x\right)-\Gamma% \left(a,x+y\right)=e^{-x}x^{a-1}\sum_{n=0}^{\infty}\frac{{\left(1-a\right)_{n}% }}{(-x)^{n}}(1-e^{-y}e_{n}(y)),$ $|y|<|x|$.
8.7.3 $\Gamma\left(a,z\right)=\Gamma\left(a\right)-\sum_{k=0}^{\infty}\frac{(-1)^{k}z% ^{a+k}}{k!(a+k)}=\Gamma\left(a\right)\left(1-z^{a}e^{-z}\sum_{k=0}^{\infty}% \frac{z^{k}}{\Gamma\left(a+k+1\right)}\right),$ $a\neq 0,-1,-2,\dots$.
8.7.4 $\gamma\left(a,x\right)=\Gamma\left(a\right)x^{\frac{1}{2}a}e^{-x}\sum_{n=0}^{% \infty}e_{n}(-1)x^{\frac{1}{2}n}I_{n+a}\left(\textstyle 2x^{1/2}\right),$ $a\neq 0,-1,-2,\dots$.
For an expansion for $\gamma\left(a,ix\right)$ in series of Bessel functions $J_{n}\left(x\right)$ that converges rapidly when $a>0$ and $x$ ($\geq 0$) is small or moderate in magnitude see Barakat (1961).