Questions tagged [beta-function]
For questions about the Beta function (also known as Euler's integral of the first kind), which is important in calculus and analysis due to its close connection to the Gamma function. It is advisable to also use the [special-functions] tag in conjunction with this tag.
634 questions
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How to derive relations between integrals over a triangle without evaluating them explicitly?
Solving a problem I found several integrals that look like
$$I(z) = \int \frac{f(x,y)}{(x y)^{1 + z}} dx dy$$
where the integral is over the triangle $\left\{(x,y) | x>0, y>0, x+y<1\right\}$.
...
- 178
8
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1
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Using the integral $\frac{1}{\binom{n}{r}} = (n+1)\int_0^1 x^r(1-x)^{n-r}\mathrm dx$
Using the integral, $$\frac{1}{\binom{n}{r}} = (n+1)\int_0^1 x^r(1-x)^{n-r} \mathrm dx,$$ I want to solve some binomial questions.
For instance, it seems, I might be unclear as to what I require. I am ...
- 579
4
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1
answer
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Representation of the beta Euler function by an infinite sum
Where can I find the derivation of this formula?
https://functions.wolfram.com/GammaBetaErf/Beta/06/03/0001/
$$B(a, b) = \sum^\infty_{k=0}\frac{(1 - b)_k}{(a + k) k!} ;\quad \Re(a) > 0 \;\text{ and ...
- 91
2
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1
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Asymptotic expansion of incomplete Beta function
Considering the incomplete beta function
$$B_x(a, b) = \int_0^x y^{a-1}(1-y)^{b-1} \:dy,$$
I am interested in deriving the following asymptotic expansion that I determined using Mathematica
$$B_{1-x}(...
2
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1
answer
124
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Closed form or representation as a hypergeometric series of $\sum _{k=1}^{\infty } \frac{x^{k-1}}{B\left(k,-\frac{k}{n}\right)}$
Working on a problem I came across this series,
\begin{equation}
\frac{1}{2}+(\frac{n-1}{n})\sum_{k=1}^{\infty}\frac{x^{-\frac{k(n-1)}{n}}}{k(k+1)B\left(k,-\frac{k}{n}\right)}
\end{equation}
Removing ...
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1
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Definite integrals involving logarithm and polylog [closed]
I am dealing with integrals in the form
$\mathcal{I} = \int_0^{1} dx \, x^\alpha \, (1-x)^\beta \, log(x)$
And
$\mathcal{J} = \int_0^{1} dx \, x^\alpha \, (1-x)^\beta \, Li_n(x)$ .
I have written the ...
6
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2
answers
383
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How to prove $I=\Gamma(\frac{1}{4})^2\frac{\sqrt{2}(\pi+4)}{8\sqrt{\pi}}-\pi-4$.
Context
Some numerical work suggests that the following integral is true:
$$\begin{align*}
I&=\int_{0}^{1}\arcsin{(x^2)}\left(\frac{\log{(\sqrt{1-x^4}+1)}}{x^2}-\frac{2x(x(1-x^4)^{1/4}-1)}{(1-x^4)^...
- 1,181
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4
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General solution to $\int_{1}^{\infty} \frac{1}{x^{2k}+1} \ dx$ via beta function
I'm trying to tackle integrals of the form
$$
\int_{1}^{\infty}
\frac{1}{x^{2k}+1}
\ dx
$$
for large $k \in \mathbb{Z}^{+}$. An easily computable general solution is desired, rather than, say, a ...
- 103
2
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2
answers
106
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Prove $ \sum_{k=0}^{2n} (-1)^k \beta\left(\frac{x+k}{2n+1}\right) = (2n+1) \beta(x) $
Prove the given identity:
$$ \sum_{k=0}^{2n} (-1)^k \beta\left(\frac{x+k}{2n+1}\right) = (2n+1) \beta(x) $$ Where $$ \beta(x) = \sum_{j=0}^{\infty} \frac{(-1)^j}{x+j} $$
The proof is more ...
user1650978
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6
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Calculating $\int_0^1 \frac{\ln(x)}{\sqrt{x - x^2}}\, dx$
I'm stuck on this problem and would appreciate any help.
I'm trying to compute the integral
$$
\int_0^1 \frac{\ln(x)}{\sqrt{x - x^2}}\, dx.
$$
To approach it, I define a parametric function
$$
F(a) = \...
- 816
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Evaluating the sum $\sum_{0}^{n}k^{i}(n-k)^{j}$ and its connection to the beta function
I want to compute the sum:
$$\sum_{0}^{n}k^{i}(n-k)^{j}$$
If I simply expand the polynomial, I get:
$=\sum_{k=0}^{R}\sum_{l=0}^{j}k^{i}\binom{j}{l}R^{l}(-1)^{j-l}k^{j-l}\\
=\sum_{l=0}^{j}\binom{j}{l}...
- 31
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1
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65
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Confusion on Beta Function Integral Representation
I found this beta function representation integral very pleasing, so I tried to work it out—but something went horribly wrong. I'm wondering if I am wrong, or if the book is. I've been working on this ...
- 1,277
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5
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How can I evaluate the close form of $\int_0^{\pi}\left(\ln\left(\sin x\right)\right)^k\text dx$
Hmm, that is what we have.
A "close form" using Beta function:
$$\begin{align}\int_0^\pi \left(\ln(\sin x)\right)^k\text dx=\frac{1}{2^k} \frac{\partial^k}{\partial a^k} \text B\left(a, \...
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1
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Finding values that make the Beta function symmetric
For finding when the median of a beta distribution is $\frac{1}{2}$, this answer says:
If a $\mathrm{Beta}(a,b)$
distribution has $a>b$ then $\mathbb P(X \le \frac12) \lt \frac12$ and
its median ...
- 2,674
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0
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Looking for an exact expression of an Euler-type integral
Reading through Caballero and Chaumont's paper 'Conditioned stable Lévy processes and the Lamperti representation', I have been trying to find an exact form of the following integral:
$$
\int_{-\infty}...
- 49