Beta

The beta function $B$ (note: $B$ is capital $\beta$ in Greek) is defined by the following formula for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1} \mathrm{d}t=\displaystyle\int_0^{\infty} \dfrac{t^{x-1}}{(1+t)^{z+y}}=2\displaystyle\int_0^{\frac{\pi}{2}} (\sin t)^{2x-1} (\cos t)^{2y-1} \mathrm{d}t.$$

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Properties

Partial derivative of beta function
Beta in terms of gamma
Beta in terms of sine and cosine
Beta as improper integral
Beta is symmetric

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

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.5 (1)$
• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $6.2.1$