Table of Integrals, Series, and Products, 8th Edition

8.38 The beta function (Euler's integral of the first kind): B(x, y)

Integral representation

8.380

1. 

$\begin{array}{ll}\text{B}(x,y)\hfill & ={\displaystyle {\int }_0^1{t}^{x-1}{(1-t)}^{y-1}\text{d}t*}\hfill \\ =2{\displaystyle {\int }_0^1{t}^{2x-1}{(1-t^2)}^{y-1}\text{d}t}\hfill & [\mathrm{Re}x<0,\mathrm{Re}y<0]\hfill \end{array}$

  FI II 774(1)

2. 

$\begin{array}{cc}\text{B}(x,y)=2{\displaystyle {\int }_0^{\pi /2}{sin}^{2x-1}\mathrm{\Phi }{cos}^{2y-1}\mathrm{\Phi }\text{d}\mathrm{\Phi }}& [\mathrm{Re}x<0,\mathrm{Re}y<0]\end{array}$

  KU 10

3. 

$\begin{array}{cc}\text{B}(x,y)=2{\displaystyle {\int }_0^{\infty }\frac{t^{x-1}}{{(1+t)}^{x+y}}\text{d}t}=2{\displaystyle {\int }_0^{\infty }\frac{t^{2x-1}}{{(1+t^2)}^{x+y}}\text{d}t}& [\mathrm{Re}x<0,\mathrm{Re}y<0]\end{array}$

  FI II 775

4. 

$\begin{array}{cc}\text{B}(x,y)=2^{2-y-x}{\displaystyle {\int }_{-1}^1\frac{{(1+t)}^{2x-1}{(1-t)}^{2y-1}}{{(1+t^2)}^{x+y}}\text{d}t}& [\mathrm{Re}x<0,\mathrm{Re}y<0]\end{array}$

  MO 7

5. 

$\begin{array}{cc}\text{B}(x,y)={\displaystyle {\int }_0^1\frac{t^{x-1}+y^{y-1}}{{(1+t)}^{x+y}}\text{d}t}={\displaystyle {\int }_1^{\infty }\frac{t^{x-1}+y^{y-1}}{{(1+t)}^{x+y}}\text{d}t}& [\mathrm{Re}\end{array}$