Questions tagged [beta-function]

For questions about the Beta function, a special function closely related to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.

304 questions
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Recursive relationship for incomplete beta function

Consider the incomplete Beta function $I_x(a, b)$ $$I_x(a, b) = \dfrac{B(x; a, b)}{B(a, b)} = \dfrac{\int_0^x t^{a-1} \left( 1-t \right)^{b-1} dt}{\int_0^1 t^{a-1} \left( 1-t \right)^{b-1} dt}.$$ ...
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A Beta function related integral

The integral $$\int \limits_{0} ^{1} \left( \frac{x}{x^2+a^2} \right)^p \frac{\log (x)}{x} dx$$ is being considered. Does anyone have an idea how to express it in terms of the Beta function? The ...
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Moments of beta distribution

I have a random variable given by $$Y = a \cdot X \;,$$ where $X$ follows a beta distribution and $a$ is a simple constant. I want to find the moments of $Y$. I am aware of the general formula for ...
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Need help calculating a definite integral

I'm trying to calculate the following integral: $$\int_0^1 e^{-\lambda(1-x)} (1-x)^{n-1} x^{k-n} dx$$ It seems like kind of a combination of Gamma and Beta function. I'm suspecting that it has ...
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Proving that $B(2\alpha,1-\alpha)\le\frac1{\alpha(1-\alpha)}$

Let $0<\alpha<\frac12$, I cannot prove that $$B(2\alpha,1-\alpha)\le\frac1{\alpha(1-\alpha)}$$ where $$B(p,q):=\int_0^1t^{p-1}(1-t)^{q-1}\,dt=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$$ is ...
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Prove the following : $\int_0^{ 1 } \frac{x^{l-1} (1-x)^{m-1}}{ (b+cx)^{l+m} }\,dx = \frac{\beta(l,m)}{(b+c)^{l}b^m}$

Prove the following : $$\int_0^{ 1 } \frac{x^{l-1} (1-x)^{m-1}}{ (b+cx)^{l+m} }\,dx = \frac{\beta(l,m)}{(b+c)^{l}b^m}$$ I found this substitution : $$x = \frac{-b}{t+c}$$ Using this, we get the ...
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Beta function with one negative and one positive arguments

Beta function defined as $B(x,y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt$ is only well defined when Re $x,$ Re $y >0$. However, according to "http://www.efunda.com/math/beta/," we can use the fact that ...
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