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GeezerVison is showing mathworld.wolfram.com — Incomplete Beta Function -- from Wolfram MathWorld

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Incomplete Beta Function

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The incomplete beta function is the generalization of the complete beta function defined by

B(z;a,b)=int_0^zu^(a-1)(1-u)^(b-1)du,
(1)

sometimes also denoted B_z(a,b). The so-called Chebyshev integral is given by

intx^p(1-x)^qdx=B(x;1+p,1+q).
(2)

The incomplete beta function is implemented in the Wolfram Language as Beta[z, a, b].

It is given in terms of hypergeometric functions by

B(z;a,b)=(z^a)/a_2F_1(a,1-b;a+1;z)
(3)
=z^aGamma(a)_2F^~_1(a,1-b;a+1;z).
(4)

It is also given by the series

B(z;a,b)=z^asum_(n=0)^infty((1-b)_n)/(n!(a+n))z^n,
(5)

where (x)_n is a Pochhammer symbol.

The incomplete beta function B(z;a,b) reduces to the usual beta function B(a,b) when z=1,

B(1;a,b)=B(a,b).
(6)

It has derivative

(dB(z;a,b))/(dz)=(1-z)^(b-1)z^(a-1)
(7)

and indefinite integral

intB(z;a,b)dz=zB(z;a,b)-B(z,a+1,b).
(8)

See also

Beta Function, Chebyshev Integral, Regularized Beta Function

Related Wolfram sites

https://functions.wolfram.com/GammaBetaErf/Beta3/, https://functions.wolfram.com/GammaBetaErf/Beta4/

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More things to try:

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References

Pearson, K. (Ed.). Tables of Incomplete Beta Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1968.

Referenced on Wolfram|Alpha

Incomplete Beta Function

Cite this as:

Weisstein, Eric W. "Incomplete Beta Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IncompleteBetaFunction.html

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