Questions tagged [gamma-function]
Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions. The Gamma function is a specific way to extend the factorial function to other values using integrals.
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Is there any similar equation for π that someone has figured out?
I'm first off sorry if my title is a little vague, but I couldn't find any other way to put it. I'm also very sorry because I am extremely bad at math, but I thought that this was interesting none-the-...
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Analytic continuation of a kind of generealized $\Gamma$ function
I wonder if analytic continuation exists for a wide class of functions. Consider a sequence of positive real numbers $u_n$ such that $u_n=O(1/n!)$ and the entire function $f(z)=\sum_{n\ge0}u_nz^n$. ...
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Clean version of inequality for $\Gamma(z)$ - known?
Let $z=x+i y,ドル $x\geq 1/2$. Is the following inequality true?
$$|\Gamma(z)|\leq (2\pi)^{1/2} |z|^{x-1/2} e^{-\pi |y|/2}$$
If you allow a fudge factor such as $e^{\frac{1}{6|z|}}$ on the right side, ...
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For what integer $n$ does $E(2/n)/K(2/n)$ reduce to Gamma functions?
Consider the complete elliptic integrals $K, E$ which are defined as follows: $$K(m)=\int_{0}^{\pi/2}\frac{dx}{\sqrt{1 - m\sin^{2}x}},,円E(m)=\int_{0}^{\pi/2}\sqrt{1-m\sin^{2}x},円dx$$
I want to know ...
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Integrating the function from 0 to infinity rootx sinx using Gamma Function
I read that you can't find the antiderivative (in closed form) of rootx sinx I tried it using gamma function of course as definite integral and got the answer rootpi/2root2. But after I asked wolfram ...
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Inverse Mellin Transform of $\Gamma(s)$
In Apostol's book "Modular Functions and Dirichlet Series in Number Theory", it states the inverse Mellin transform of the gamma function:
$$e^{-x} = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\...
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Generalisation of $\int_0^\infty \cos(x^2)\ln(x)dx .$
In the post, it is found that
$$\int_0^\infty \cos(x^2)\ln(x)dx=-\frac 18\sqrt{\frac {\pi}2}\Bigr(\pi+4\log 2+2\gamma\Bigr),$$
its generalisation $$I(\alpha)=\int_0^\infty \cos(x^\alpha)\ln(x)dx= \...
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Connection between Watson Triple Integral and Gauss Constant?
I noticed that the first of Watson's triple integrals is equal to 2ドル G^2,ドル where $G$ is Gauss's Constant
$$
I_1 = \frac{1}{\pi^3} \int_0^\pi \int_0^\pi \int_0^\pi \frac{du\ dv\ dw}{1 - \cos u \cos v \...