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.

305 questions
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Integral of the incomplete beta function

Does anyone know how to evaluate $$I(a,b) = \int_0^1 B_t(a,b) dt$$? Here $B_t(a,b)$ is the incomplete Beta function, defined as $$B_t(a,b) = \int_0^t x^{a-1}(1-x)^{b-1} dx$$ Of course $I(a,b)$ ...
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Using the Eulerian integrals evaluate $\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x$

The question asks to evaluate the integral: $$\int_0^\infty \frac{\ln^2{x}}{1+x^4} \mathrm{d}x.$$ I have tried a few substitutions but am not getting anywhere. Thanks in advance!
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Double integral involving beta functions

I want to prove the following result: $$\int_0^1 \int_0^1 {f(xy)(1-x)^{p-1}y^p(1-y)^{q-1}} \mathrm{d}x \, \mathrm{d}y=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)} \int_0^1 {f(t)(1-t)^{p+q-1}} \mathrm{d}t.$$...
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How to Integrate $\int^{\pi/2}_{0} x \ln(\cos x) \sqrt{\tan x}\,dx$

Evaluate $$\displaystyle \int^{\frac{\pi}{2}}_{0} x \ln(\cos x) \sqrt{\tan x}dx$$ Unfortunately, I have no idea on how to integrate this and thus cannot provide any inputs on my own. The only ...
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Beta function of the conjugate arguments

How to simplify (integrate) $$\text{B}\left(\dfrac{m}{2}+ix, \dfrac{m}{2}-ix\right)$$ when $m\in\mathbb{N}$? eg. when $m=1$ WolframAlpha simplifies the expression to $\pi \ \text{sech} (\pi x)$
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Derivation of continued fraction for the incomplete beta function?

Where can I find a derivation for this continued fraction representation of the incomplete beta function: http://dlmf.nist.gov/8.17#v? I would like to have a reference to the papers where this ...