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.

301 questions
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Closed form for this integral with Beta function $\int_0^\infty x \mathrm B (x,x)~dx$

Is it possible to find the closed form of $$\int_0^\infty x \mathrm B (x,x)~dx=2.44333\dots$$ This integral converges because for small $x$ Beta function behaves like $2/x$. Using the integral ...
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An integral involving beta function

For $-1 \leq \theta \leq 1$ and $\nu > -1/2$, prove that the function $$f(x; \theta,\nu) = \frac{(1-x^2)^{\nu -1/2}} {(1-2\theta x +\theta^2)^\nu B(\nu+1/2,1/2)}$$ is a valid probability density ...
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Is there any special property about incomplete beta function $B(x;1-x,x)$?

guys, I encountered a function like $$f(x) = B(x;1-x,x)$$ where $B(\cdot)$ is the incomplete beta function and input $0 < x < 1$ is some positive small real value close to zero . I want to ...
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I like to know beta function`s property related with my question

Fact 1. ($B$ is beta-function) $$\int_{0}^{1}\frac{z^{-(n+1)}}{(1+\frac{1}{z})^{2n}}=\frac{1}{2}B(n,n)$$ I can find above fact by using MATLAB. But i like to show above fact using beta function ...
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Compute $\int_0^1x^m(1-x^n)^pdx$

Compute $\int_0^1x^m(1-x^n)^pdx$ Hint in question says, express in terms of gamma function, which I don't see how. I can put $x=\frac{1}{e^t}$ and make limit from $0$ to infinity, as in gamma ...
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Parameterizing monotonic functions

I am trying to design a function $f_{\alpha} : \left[0,1\right] \rightarrow [0,+\inf)$ that looks like the following: It should have the following properties: Its integral should be 1 for any value ...
Evaluate $\int_0^1\frac{\ln{\left(1-kt^2(1-t)+\frac{t^4(1-t)^2}{4}\right)}}{t}dt$ , where $k=\cos1$
I want to find the value of $\displaystyle\sum_{n=1}^\infty \dfrac{\cos n}{n^22^n\binom{3n}{n}}.$ Since $\displaystyle\dfrac{1}{\binom{3n}{n}}=2n\beta(2n,n+1)=2n\int_0^1t^{2n-1}(1-t)^ndt$, \$\...