All Questions
Tagged with closed-form gamma-function
109
questions
1
vote
2
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345
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Closed form solution for $\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx$
I am trying to calculate Fourier series coefficients (by hand) and the integrals I need to solve are of the following type
$$I(N,M,n,m)=\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx,$$
in ...
- 15.9k
14
votes
1
answer
1k
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Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$
Mathematica gives the following. But how?!
$$\small{\int_0^1 \dfrac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(
9\Gamma\left(\tfrac{3}{4}\right)^2{}_4F_3\left( \begin{array}{c}\...
- 27k
2
votes
3
answers
130
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Does this function have a closed form?
$$k_a(x)=\sum^\infty_{n=0}{\frac{n^2}{(n+x)!}}$$
$$k_b(x)=\sum^\infty_{n=0}{\frac{(n+x)^2}{n!}}$$
I noticed these functions closely relate to $e$. By looking at them I was able to determain a closed ...
- 4,472
9
votes
1
answer
448
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Simple closed form for $\int_0^\infty\frac{1}{\sqrt{x^2+x}\sqrt[4]{8x^2+8x+1}}\;dx$
Some time ago, I used a fairly formal method (in the second sense of this answer) to derive the following integral, and am wondering whether it is correct or not:
$$\int_0^\infty\frac{1}{\sqrt{x^2+...
- 3,343
13
votes
2
answers
368
views
Prove known closed form for $\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx$
I know that the following identity is correct, but I would love to see a derivation:
$$\int_0^\infty e^{-x}I_0\left(\frac{x}{3}\right)^3\;dx=\frac{\sqrt{6}}{32\pi^3}\Gamma\left(\frac{1}{24}\right)\...
- 3,343
28
votes
3
answers
1k
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Interesting closed form for $\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta$
Some time ago I used a formal approach to derive the following identity:
$$\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta=\frac{3^{\frac{1}{12}}\pi\...
- 3,343
22
votes
2
answers
2k
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Closed form (or an ODE) for the integral $\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$
Is there a closed form for: $$I(p)=\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$$
The integral has a number of nice properties:
$$I(p)=I(-p)$$
$$I(p)=2\int_0^1 \frac{1+z^2}{1+z^4} \...
- 31.7k
13
votes
3
answers
537
views
Is $\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)^2\,\Gamma(\sigma-xi)^2 \,dx = \frac{\Gamma(2\,\sigma)^4}{\Gamma(4\,\sigma)}$?
Using the approach from the answer to this question, it can be shown that for $\sigma \in \mathbb{C}, x\in \mathbb{R}$:
$$\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)\,\Gamma(\sigma-xi) \,dx =\...
- 3,191
3
votes
0
answers
134
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Is there a known transformation between $_2F_1\big(\tfrac12,\tfrac12;1;z\big)$ and $_2F_1\big(\tfrac12,\tfrac12;1;z^2\big)$?
In this post, the OP seeks a closed-form for,
$$A=\,_2F_1\big(\tfrac12,\tfrac12;1;\tfrac19\big)=1.02966\dots$$
Using the transformation,
$$\,_2F_1\big(\tfrac12,\tfrac12;1;z\big) = \tfrac2{1+\sqrt{1-z}}...
- 54.9k
21
votes
3
answers
1k
views
On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $
I. $k=4n+3.\;$ From this post, one knows that $$\sum_{n=1}^\infty \frac{n^{3}}{e^{2n\pi}-1} = \frac{\Gamma\big(\tfrac{1}{4}\big)^8}{2^{10}\cdot5\,\pi^6}-\frac{1}{240}$$ and a Mathematica session ...
- 54.9k
0
votes
0
answers
196
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Closed form for this sum involving the lower incomplete gamma function?
Can this sum be written in simpler terms?
$$\sum_{k=0}^\infty \frac{1}{z-k} \cdot \frac{\gamma(k,-\log x)}{\Gamma(k)}$$
(where $\gamma(k,-\log x)$ is the lower incomplete gamma function)
I'm pretty ...
- 987
2
votes
1
answer
232
views
Approximation of the factorial function
I'm using the term factorial function as $\gamma(x+1)$ on the sense that I'm taking all real number in count.
I have seen many approximations of the factorial function for positive values, for ...
- 5,166
14
votes
2
answers
400
views
Integral ${\large\int}_0^1\frac{dx}{(1+x^{\sqrt2})^{\sqrt2}}$
Mathematica claims that
$${\large\int}_0^1\!\!\frac{dx}{(1+x^{\sqrt2})^{\sqrt2}}=\frac{\sqrt\pi}{2^{\sqrt2}\sqrt2}\cdot\frac{\Gamma\left(\frac1{\sqrt2}\right)}{\Gamma\left(\frac12+\frac1{\sqrt2}\right)...
- 47.3k
1
vote
1
answer
90
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How do you calculate the case $\lambda=2$ of this identity related to $\int_0^\infty \frac{x^{\lambda(s-1)}}{e^x+1}dx$?
Inspired in an integral representation for the Dirichlet Eta function (that is the alternating series of the Riemann Zeta function) I've calculated some integrals using Wolfram Alpha.
Example 1. ...
user243301
2
votes
1
answer
143
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Proof that $\int_0^1\frac{(-\log u)^s}{u^s}du=\frac{\Gamma(s+1)}{(1-s)^{s+1}}$ for $|\Re s|<1$
After I've read an identity involving an integral related with special functions, I've consider a different integral by trials asking to Wolfram Alpha online calculator
Example For the code int_0^1 ...
user243301