Questions tagged [gamma-function]
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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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Find derivative of such complicated expression.
I am trying to obtain the derivative of the following expression with respect to $x$, but not getting it correctly. $$F = \biggl\{\frac{\exp(\delta){\sigma_5}}{\sqrt{\alpha^2\sigma^2_2\sigma^2_4}}\... derivatives gamma-function hypergeometric-function- 61
How to solve such complicated integration involving product of two lower incomplete Gamma function?
I am trying to solve the following integration but not getting it correctly. $$P = \int_0^\infty \left(\gamma\left(1+b,\frac{t(1+x)-1}{A\sigma^2_1}\right)\right)^N\cdot\frac{\exp\left(\frac{-x}{A_1\... definite-integrals gamma-function hypergeometric-function- 61
$-2(-1)^n\zeta(n)=\lim_{m\to\infty}m^n\left(\displaystyle\prod_{k=1}^n\Gamma\left(1+\frac{\zeta_n^k}{m}\right)^{-2}-1\right).$
Theorem Let $\zeta_n=e^{2\pi i/n}$ be the $n$-th root of unity, then $$-2(-1)^n\zeta(n)=\lim_{m\to\infty}m^n\left(\displaystyle\prod_{k=1}^n\Gamma\left(1+\frac{\zeta_n^k}{m}\right)^{-2}-1\right).$$ ... gamma-function riemann-zeta- 9,883
Complex Parts of the Gamma function
The Gamma Function is defined as the following integral: $$\Gamma(z) = \int_0^\infty x^{z-1}e^{-x}\, \text{d}x$$ For $z \in \mathbb{C}$, and $\text{Re}(z)$ is not a non-negative integer. If we let $z =... complex-analysis numerical-methods gamma-function- 347
I'm trying to turn this integral into a Gamma function but it seems impossible!
Trying to find the principal value of this integral by turning it into a Gamma function but it doesn't seem to have a closed form: $\int_a^ \infty dx\frac{x^2}{\sqrt{x^2+a^2}}\ \frac{1}{e^x-1}\ \frac{... gamma-function cauchy-principal-value- 11
Lower bound on the upper incomplete Gamma function
The upper incomplete Gamma function is defined as $\Gamma(a,x):=\int_x^\infty t^{a-1}e^{-t}dt$ for any $a\in \mathbb{R}$ and $x\in \mathbb{R}^+$. I am trying to find an exact, preferably tight, lower ... real-analysis gamma-function- 3,156
Minimum of the Gamma function
Working on one of other question Show the inequality $\frac{\sqrt{\pi}}{2}<\left(\pi-e\right)!$ I found : $$\frac{\left(\pi-e+\frac{1}{2}\right)}{2}\simeq x_{min}=0.4616\cdots$$ Where $x_{min}$ ... power-series maxima-minima gamma-function- 3,390
Let $a>0$. Prove the improper integral $\int_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx$ converges for $k>2$
There is a hint which says $\left|\int\limits_0^{\infty} \cos\left\{{a\over2}\left(x+{1\over x}\right)\right\}x^{k-2}\ dx\right|\le C a^{k-2}$ where C is some constant. I somehow feel that I need to ... real-analysis integration analysis improper-integrals gamma-function- 371
How do software packages compute the gamma probability density for large $\alpha$?
Consider the gamma density parameterized in terms of shape $(\alpha)$ and rate $(\beta)$: $$ f(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}\mathbf 1_{x>0}. $$ Direct computation ... gamma-function computational-mathematics gamma-distribution- 4,071
Neat function $I(x,y)= \sum\limits_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg| $. Closed form?
Consider the following function: $$ I(x,y)=\sum_{n=1}^{\infty}\bigg|\int_0^1 \frac{1}{t~(\log t)^y}\exp\bigg(\frac{n^x}{\log t}\bigg) ~dt~\bigg|$$ For $x=0$ and letting $y$ vary we get the Gamma ... integration special-functions closed-form gamma-function riemann-zeta- 801
Looking for $B$ such that $\Gamma(ax+b) \prod_{i=1}^n (ax+e_i)\sim \Gamma(ax+B)$
Quite often, I need to solve for $x$ $$y=\Gamma(ax+b)\prod_{i=1}^n (c_ix+d_i)\tag 1$$which, numerically does not make much problems looking for the zero of function $$f(x)=\log\Big[\Gamma(ax+b)\Big]+\... approximation gamma-function newton-raphson lambert-w- 215k
Definite Integral of $\text{exp}(-(\sum_i |x_i|^a)^b)$
I would like to solve: $$\int_{\mathbf{R}^n} \text{exp}\left(-\left(\sum_i |x_i|^a\right)^b\right)\;dx_1...dx_n$$ I have no idea how to tackle this integral. For the special case of $a=1$ I obtained ... multivariable-calculus definite-integrals normal-distribution gamma-function gaussian-integral- 13
Prove $(n-1)! + 1 = n^2$ has only one integer solution [duplicate]
Prove $(n-1)! + 1 = n^2$ has only one integer solution, namely $5$. I think that we can use the derivate of the gamma function to say that the LHS is growing more than the RHS from $n=5$ onwards, so ... calculus algebra-precalculus elementary-number-theory gamma-function digamma-function- 123
how to interpret this derivation of beta function as a ratio of gamma functions?
The derivation is here (screenshot included in case the link changes in the future). The parts that confuse me are Why is it important that we state the determinant of the Jacobian of the ... integration gamma-function beta-function- 603
The integral $\int_{0}^{\pi/2} \tan^p x~dx$
The integral $I=\int_{0}^{\pi/2} \tan^p x~ dx$ is positive and convergent for $0<p<1$. However, the Beta integral along with the property of Gamma function yields the integral \begin{align}I&... definite-integrals gamma-function beta-function- 38k
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