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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Is a Beta distribution a continuous version of the Binomial Theorem?

The visual appearance of the PDF for a Beta distribution resembles that for the terms in the Binomial Theorem. Is the former a continuous variant of the discrete terms of the latter? Are they ...
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Peculiar (convergent?) definite integral

I have been trying to calculate the integral: $$\int_1^{\infty} \left(\frac{x^2}{\sqrt{x^4-1}}-1\right)dx$$ A hint is to multiply the whole integral by $x^{\lambda}$, calculate the two terms ...
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Beta function in Philip J. Davis׳ Essay

This question is about equation number (4) in Philip J. Davis’ Essay titled "LEONHARD EULER'S INTEGRAL: A HISTORICAL PROFILE OF THE GAMMA FUNCTION". In there it is stated by the author "Euler ...
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conditonal distribution question

For conditional distribution $$f_{X|Y}(x|y) = \frac{f(x,y)}{f_Y(y)}$$ this is the basic definition I know about conditional distribution Consider n + m trials having a common probability of ...
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Calculate $\int_{0}^{\pi} \frac{dx}{\sqrt{3-\cos(x)}}$ [closed]

$$\int_{0}^{\pi} \frac{dx}{\sqrt{3-\cos(x)}}$$ I need to calculate this using Beta \ Gamma functions. I have tried the substitution $2 +\cos(x) = t$
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Integral $\int_{a}^s\int_{a}^u\frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu$

Is it possible to get a closed form for $$\int_{a}^s\int_{a}^u\frac{1}{\sqrt{s-u}\sqrt{u-v}\sqrt{v-a}}dvdu\quad?$$ If we look at the simple integral it is related to Beta function, but for the ...
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What are the poles and zeros of the Euler Beta function?

For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ does the Euler Beta function $B(x,y)$ equal zero? For what pairs of complex values $(x,y) \in \mathbb{C} \times \mathbb{C}$ ...
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Solution of $\int_x^1y^{a-1}\left(1-y\right)^{b-1}dy = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$

When is $f=g$ on $(0,1)$ for $f = \int_x^1y^{a-1}\left(1-y\right)^{b-1}dy$ $g = \left(2\frac{x+1}{x+2}\right)x^{a}\left(1-x\right)^{b-1}$ Let me show their graphs. They are small, so I multiplied ...
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Incomplete Beta function

I am looking for an approximation/bound for the incomplete Beta function $B_z(a,b)$ when $z\to0$. I know the Taylor expansion would help. However, I need a power series in $z^n$ (the exponential is an ...
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Seeking methods to solve: $\int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt$

Seeking Methods to solve the following two definite integrals: \begin{equation} I_(n) = \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt \qquad J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt \end{equation}...
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How to show: $\psi^{(0)}\left(\frac{1}{n}\right) - \psi^{(0)}\left(1 - \frac{1}{n}\right) = -\pi\cot\left(\frac{\pi}{n}\right)$ [duplicate]

Based on a result I found recently and in conjunction with methods I've observed on MSE I was able to show that: \begin{equation} \int_0^\infty \frac{ \ln(t)}{t^n + 1}\:dt = -\frac{\pi^2}{n^2} \...
Estimate the probability of success Suppose I send 10 tasks to my machine. 6 out of 10 tasks success, and 4 failed. These outcomes is summarized by $X$ as a binary variable, 1 is task success, and 0 ...
I would like to prove the left side to right hand side which is in form of hypergeometeric function. Looking for your hints, suggestions and solultions.  \alpha_{1} \int_{0}^{1} (1-z)^{\alpha_{1}+\...