Euler's beta function

The beta function , also Euler integral first type (after Leonhard Euler ) is a mathematical function of two complex numbers , with is called. Its definition is: ${\displaystyle \mathrm {B} }$

Beta function. The positive real parts of x and y lie in the plane

${\displaystyle \mathrm {B} (x,y)=\int \limits _{0}^{1}t^{x-1}(1-t)^{y-1}\,\mathrm {d} t,}$

where and must have a positive real part . ${\displaystyle x}$ ${\displaystyle y}$

The beta function occurs, among other things, in the beta distribution .

General

For fixed (or ) is a meromorphic function of (or ), and the symmetry relation applies to the function ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle \mathrm {B} }$ ${\displaystyle y}$ ${\displaystyle x}$

${\displaystyle \mathrm {B} (x,y)=\mathrm {B} (y,x)}$.

The following additional integral representations exist for the beta function with and (the first representation results from the substitution ) ${\displaystyle \mathrm {Re} \,x>0}$ ${\displaystyle \mathrm {Re} \,y>0}$ ${\displaystyle u={\tfrac {t}{1-t}}}$

${\displaystyle {\begin{aligned}\mathrm {B} (x,y)&{}=\int \limits _{0}^{\infty }{\frac {t^{x-1}}{{(1+t)}^{x+y}}}\,\mathrm {d} t\\&{}=2\int \limits _{0}^{\frac {\pi }{2}}\sin ^{2y-1}(t)\cos ^{2x-1}(t)\mathrm {d} t.\end{aligned}}}$

The main result of the theory of beta function is identity

${\displaystyle \mathrm {B} (x,y)={\frac {\Gamma (x)\cdot \Gamma (y)}{\Gamma (x+y)}}}$

where denotes Euler's gamma function . This representation also shows that the analytical continuation of the beta function has poles exactly along and for whole numbers . ${\displaystyle \Gamma }$ ${\displaystyle x=k}$ ${\displaystyle y=k}$ ${\displaystyle k\leq 0}$

Theodor Schneider showed in 1940 that the number is transcendent for all rational, non-integer numbers . ${\displaystyle \mathrm {B} (x,y)}$ ${\displaystyle x,y}$

Representations

The beta function has many other representations such as:

${\displaystyle \mathrm {B} (x,y)=2\int _{0}^{\pi /2}(\sin \theta )^{2x-1}(\cos \theta )^{2y-1}\,d\theta ,\qquad {\textrm {Re}}(x)>0,\ {\textrm {Re}}(y)>0}$

${\displaystyle \mathrm {B} (x,y)=\int _{0}^{\infty }{\dfrac {t^{x-1}}{(1+t)^{x+y}}}\,dt,\qquad {\textrm {Re}}(x)>0,\ {\textrm {Re}}(y)>0}$

${\displaystyle \mathrm {B} (x,y)=\sum _{n=0}^{\infty }{\dfrac {n-y \choose n}{x+n}},}$

${\displaystyle \mathrm {B} (x,y)={\frac {x+y}{xy}}\prod _{n=1}^{\infty }\left(1+{\dfrac {xy}{n(x+y+n)}}\right)^{-1},}$

${\displaystyle \mathrm {B} (x,y)\cdot \mathrm {B} (x+y,1-y)={\dfrac {\pi }{x\sin(\pi y)}},}$

${\displaystyle \mathrm {B} (x,y)={\dfrac {1}{y}}\sum _{n=0}^{\infty }(-1)^{n}{\dfrac {y^{n+1}}{n!(x+n)}}}$

The beta function can be used to define the binomial coefficients by adjusting the indices :

${\displaystyle {n \choose k}={\frac {1}{(n+1)\mathrm {B} (n-k+1,k+1)}}.}$

With the representation for the gamma function one arrives for integer positive and on: ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle \mathrm {B} (x,y)={\dfrac {(x-1)!(y-1)!}{(x+y-1)!}}}$.

Derivation

The derivation is given by

${\displaystyle {\partial \over \partial x}\mathrm {B} (x,y)=\mathrm {B} (x,y)\left({\Gamma '(x) \over \Gamma (x)}-{\Gamma '(x+y) \over \Gamma (x+y)}\right)=\mathrm {B} (x,y)(\psi (x)-\psi (x+y))}$

where is the digamma function . ${\displaystyle \psi (x)}$

Web links

• Eric W. Weisstein: Beta Function , Regularized Beta Function , Incomplete Beta Function in MathWorld (English)
• Beta function. Evaluation at functions.wolfram.com (English)

Individual evidence

1. ^ Theodor Schneider : On the theory of Abelian functions and integrals (January 22, 1940), Journal for pure and applied mathematics 183, 1941, pp. 110–128 (at the GDZ: [1] )