Beta
The beta function $B$ (note: $B$ is capital $\beta$ in Greek) is defined by the following formula for $\mathrm{Re}(x)>0$ and $\mathrm{Re}(y)>0$: $$B(x,y)=\displaystyle\int_0^1 t^{x-1}(1-t)^{y-1} \mathrm{d}t.$$
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
B(x,y)=integral (t^(x-1)+t^(y-1))(1+t)^(-x-y) dt
B(x,y+1)=(y/x)B(x+1,y)
B(x,y+1)=(y/(x+y))B(x,y)
B(x,y)B(x+y,z)=B(y,z)B(y+z,x)
B(x,y)B(x+y,z)=B(z,x)B(x+z,y)
B(x,y)B(x+y,z)B(x+y+z,u)=Gamma(x)Gamma(y)Gamma(z)Gamma(u)/Gamma(x+y+z+u)
1/B(n,m)=m((n+m-1) choose (n-1))
1/B(n,m)=n((n+m-1) choose (m-1))
B(x,y)=2^(1-x-y)integral (1+t)^(x-1)(1-t)^(y-1)+(1+t)^(y-1)(1-t)^(x-1) dt
Integral t^(x-1)(1-t)^(y-1)(1+bt)^(-x-y)dt = (1+b)^(-x)B(x,y)
Integral t^(x-1)(1+bt)^(-x-y) dt = b^(-x) B(x,y)
Integral (t-b)^(x-1)(a-t)^(y-1)dt=(a-b)^(x+y-1)B(x,y)
Integral of (t-b)^(x-1)(a-t)^(y-1)/(t-x)^(x+y) dt=(a-b)^(x+y-1)/((a-c)^x(b-c)^y) B(x,y)
Integral of (t-b)^(x-1)(a-t)^(y-1)/(c-t)^(x+y) dt = (a-b)^(x+y-1)/((c-a)^x (c-b)^y) B(x,y)
Integral of (1+bt^z)^(-y)t^x dt = (1/z)*b^(-(x+1)/z) B((x+1)/z,y-(x+1)/z)
Integral of t^(x-1)(1-t^z)^(y-1) dt=(1/z)B(x/z,y)
Integral of (1+t)^(2x-1)(1-t)^(2y-1)(1+t^2)^(-x-y)dt=2^(x+y-2)B(x,y)
Videos
Beta integral function - basic identity (5 December 2011)
Beta Function - Gamma Function Relation Part 1 (5 December 2011)
Beta Function - Gamma Function Relation Part 2 (5 December 2011)
Beta function - Part 1 (14 February 2012)
Mod-04 Lec-09 Analytic continuation and the gamma function (Part I) (3 June 2014)
Gamma function - Part 10 - Beta function (31 July 2012)
Beta function (19 September 2012)
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Beta Integral: Even Powers Of Sine Function (26 December 2012)
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): $6.2.1$
