Beta

The beta function $B$ (note: $B$ is capital $\beta$ in Greek) is defined by the formula $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1}dt.$$

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Properties

Theorem: $B(x,y)=B(y,x)$

Proof: █

Theorem: (i) $B(x+1,y)=\dfrac{x}{x+y} B(x,y)$
(ii) $B(x,y+1)=\dfrac{y}{x+y}B(x,y)$

Proof: █

Partial derivative of beta function
Beta in terms of gamma
Beta in terms of sine and cosine

Videos

Beta function - Part 1
Beta function
Beta integral function - basic identity
Gamma function - Part 10 - Beta function
Mod-04 Lec-09 Analytic continuation and the gamma function (Part I)
Gamma Function, Transformation of Gamma Function, Beta Function, Transformation of Beta Function
Beta Function - Gamma Function Relation Part 1
Beta Function - Gamma Function Relation Part 2
Beta Integral: Even Powers Of Sine Function

References

Bell. Special Functions
Special functions by Leon Hall