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

#### Integral representation

8.380

1.

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

FI II 774(1)

2.

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

KU 10

3.

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

FI II 775

4.

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

MO 7

5.

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

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