DLMF: GA.9 Integral Representations

§GA.9. Integral Representations

Sources:

Olver(1997)(Chapter 2, §§1 and 2), Temme(1996)(Chapter 3), Whittaker and Watson(1927)(§§12.3–12.32 and §13.6). (GA.9.3) follows from (GA.2.1) by a change of variables. (GA.9.8) and (GA.9.9) follow from (GA.9.6) and (GA.9.7). (GA.9.15) and (GA.9.17) are the differentiated forms of (GA.9.10) and (GA.9.11).

§GA.9(i)Gamma Function
§GA.9(ii)Psi Function and Euler's Constant

§GA.9(i). Gamma Function

Notes:

GA.9.1

GA.9.2

Olver(1997)(pp. 37–38)

Temme(1996)(pp. 43–45 and 77)

Whittaker and Watson(1927)(pp. 251 and 277)

Keywords:

gamma function

GA.9.1 $\frac{1}{\mu}\Gamma\!\left(\frac{\nu}{\mu}\right)\frac{1}{z^{{\nu/\mu}}}=\int _{0}^{\infty}\mathrm{exp}\!\left(-zt^{\mu}\right)t^{{\nu-1}}dt,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Referenced by:: §GA.9(i)
Encodings:: pMathML, png, TeX

$\realpart{\nu}>0$ , $\mu>0$ , and $\realpart{z}>0$ . (The fractional powers have their principal values.)

Hankel's Loop Integral

Keywords:: gamma function, Hankel's loop integral

GA.9.2 $\frac{1}{\Gamma\!\left(z\right)}=\frac{1}{2\pi i}\int _{{-\infty}}^{{(0+)}}e^{t}t^{{-z}}dt,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.1.4 (in a slightly different form.)
Referenced by:: §GA.9(i)
Encodings:: pMathML, png, TeX

where the contour begins at $-\infty$ , circles the origin once in the positive direction, and returns to $-\infty$ . $t^{{-z}}$ has its principal value where $t$ crosses the positive real axis, and is continuous. See Figure GA.9.1.

GA.9.1 $t$ -plane. Contour for Hankel's loop integral.

Magnify

Referenced by:: §GA.9(i)
Encodings:: eps, png

GA.9.3 $c^{{-z}}\Gamma\!\left(z\right)=\int _{{-\infty}}^{\infty}|t|^{{2z-1}}e^{{-ct^{2}}}dt,$ $c>0$ , $\realpart{z}>0$ ,

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Referenced by:: §GA.9(i), §GA.9
Encodings:: pMathML, png, TeX

where the path is the real axis.

GA.9.4 $\Gamma\!\left(z\right)=\int _{1}^{\infty}t^{{z-1}}e^{{-t}}dt+\sum _{{k=0}}^{\infty}\frac{(-1)^{k}}{(z+k)k!},$ $z\neq 0,-1,-2,\dots$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §GA.9(i)
Encodings:: pMathML, png, TeX

GA.9.5 $\Gamma\!\left(z\right)=\int _{0}^{\infty}t^{{z-1}}\left(e^{{-t}}-\sum _{{k=0}}^{n}\frac{(-1)^{k}t^{k}}{k!}\right)dt,$ $-n-1<\realpart{z}<-n$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $n$ : nonnegative integer, $k$ : nonnegative integer and $z$ : complex variable
Encodings:: pMathML, png, TeX

GA.9.6 $\Gamma\!\left(z\right)\mathrm{cos}\!\left(\tfrac{1}{2}\pi z\right)=\int _{0}^{\infty}t^{{z-1}}\mathrm{cos}tdt,$ $0<\realpart{z}<1$ ,

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Referenced by:: §GA.9(i), §GA.9
Encodings:: pMathML, png, TeX

GA.9.7 $\Gamma\!\left(z\right)\mathrm{sin}\!\left(\tfrac{1}{2}\pi z\right)=\int _{0}^{\infty}t^{{z-1}}\mathrm{sin}tdt,$ $-1<\realpart{z}<1$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Referenced by:: §GA.9(i), §GA.9
Encodings:: pMathML, png, TeX

GA.9.8 $\Gamma\!\left(1+\frac{1}{n}\right)\mathrm{cos}\!\left(\frac{\pi}{2n}\right)=\int _{0}^{\infty}\mathrm{cos}\!\left(t^{n}\right)dt,$ $n=2,3,4,\dots$ ,

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $n$ : nonnegative integer
Referenced by:: §GA.9(i), §GA.9
Encodings:: pMathML, png, TeX

GA.9.9 $\Gamma\!\left(1+\frac{1}{n}\right)\mathrm{sin}\!\left(\frac{\pi}{2n}\right)=\int _{0}^{\infty}\mathrm{sin}\!\left(t^{n}\right)dt,$ $n=2,3,4,\dots$ .

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $n$ : nonnegative integer
Referenced by:: §GA.9(i), §GA.9
Encodings:: pMathML, png, TeX

Binet's Formula

Keywords:: Binet's formula, gamma function

GA.9.10 $\mathrm{ln}\Gamma\!\left(z\right)=\left(z-\tfrac{1}{2}\right)\mathrm{ln}z-z+\tfrac{1}{2}\mathrm{ln}\!\left(2\pi\right)+2\int _{0}^{\infty}\frac{\mathrm{arctan}\!\left(t/z\right)}{e^{{2\pi t}}-1}dt,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
A&S Ref:: 6.1.50
Referenced by:: §GA.9(i), §GA.9(ii), §GA.9
Encodings:: pMathML, png, TeX

where $|\mathrm{ph}z|<\pi/2$ and the inverse tangent has its principal value.

GA.9.11 $\mathrm{ln}\Gamma\!\left(z+1\right)=-\EulerConstant z-\frac{1}{2\pi i}\int _{{-c-\infty i}}^{{-c+\infty i}}\frac{\pi z^{{-s}}}{s\mathrm{sin}\!\left(\pi s\right)}\zeta\!\left(-s\right)ds,$

Notations:: $\Gamma\!\left(z\right)$ : Gamma function, $\EulerConstant$ : Euler's constant, $s$ : real or complex variable and $z$ : complex variable
Referenced by:: §GA.9(i), §GA.9(ii), §GA.9
Encodings:: pMathML, png, TeX

where $|\mathrm{ph}z|\le\pi-\delta$ ( $<\pi$ ), $1<c<2$ , and $\zeta\!\left(s\right)$ is as in Chapter ZE.

For additional representations see Whittaker and Watson(1927)(§§12.31–12.32).

§GA.9(ii). Psi Function and Euler's Constant

Notes:

Olver(1997)(pp. 39–40)

Temme(1996)(pp. 55–56)

Whittaker and Watson(1927)(pp. 246–251)

Keywords:

Euler's constant, psi function

For $\realpart{z}>0$ ,

GA.9.12 $\psi\!\left(z\right)=\int _{0}^{\infty}\left(\frac{e^{{-t}}}{t}-\frac{e^{{-zt}}}{1-e^{{-t}}}\right)dt,$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.21
Encodings:: pMathML, png, TeX

GA.9.13 $\psi\!\left(z\right)=\mathrm{ln}z+\int _{0}^{\infty}\left(\frac{1}{t}-\frac{1}{1-e^{{-t}}}\right)e^{{-tz}}dt,$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
Encodings:: pMathML, png, TeX

GA.9.14 $\psi\!\left(z\right)=\int _{0}^{\infty}\left(e^{{-t}}-\frac{1}{(1+t)^{z}}\right)\frac{dt}{t},$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.21
Encodings:: pMathML, png, TeX

GA.9.15 $\psi\!\left(z\right)=\mathrm{ln}z-\frac{1}{2z}-2\int _{0}^{\infty}\frac{tdt}{(t^{2}+z^{2})(e^{{2\pi t}}-1)}.$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function and $z$ : complex variable
A&S Ref:: 6.3.21
Referenced by:: §GA.9(ii), §GA.9
Encodings:: pMathML, png, TeX

GA.9.16 $\psi\!\left(z\right)+\EulerConstant=\int _{0}^{\infty}\frac{e^{{-t}}-e^{{-zt}}}{1-e^{{-t}}}dt=\int _{0}^{1}\frac{1-t^{{z-1}}}{1-t}dt.$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function, $\EulerConstant$ : Euler's constant and $z$ : complex variable
A&S Ref:: 6.3.22
Encodings:: pMathML, png, TeX

GA.9.17 $\psi\!\left(z+1\right)=-\EulerConstant+\frac{1}{2\pi i}\int _{{-c-\infty i}}^{{-c+\infty i}}\frac{\pi z^{{-s-1}}}{\mathrm{sin}\!\left(\pi s\right)}\zeta\!\left(-s\right)ds,$

Notations:: $\psi\!\left(z\right)$ : Psi or digamma function, $\EulerConstant$ : Euler's constant, $s$ : real or complex variable and $z$ : complex variable
Referenced by:: §GA.9(ii), §GA.9
Encodings:: pMathML, png, TeX

where $|\mathrm{ph}z|\le\pi-\delta(<\pi)$ and $1<c<2$ .

GA.9.18 $\EulerConstant=-\int _{0}^{\infty}e^{{-t}}\mathrm{ln}tdt=\int _{0}^{\infty}\left(\frac{1}{1+t}-e^{{-t}}\right)\frac{dt}{t}=\int _{0}^{1}(1-e^{{-t}})\frac{dt}{t}-\int _{1}^{\infty}e^{{-t}}\frac{dt}{t}=\int _{0}^{\infty}\left(\frac{e^{{-t}}}{1-e^{{-t}}}-\frac{e^{{-t}}}{t}\right)dt.$

Notations:: $\EulerConstant$ : Euler's constant
A&S Ref:: 6.3.22
Encodings:: pMathML, png, TeX

§GA.9. Integral Representations

Contents

§GA.9(i). Gamma Function

Hankel's Loop Integral

Binet's Formula

§GA.9(ii). Psi Function and Euler's Constant