# 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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### Why is gamma function defined such that $\Gamma (n)=(n-1)!$ rather then $\Gamma(n)=n!$ [duplicate]

Why is gamma function defined such that $\Gamma (n)=(n-1)!$ rather then $\Gamma(n)=n!$, The latter ssems far more logical.
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### Prove that $\gamma<\int_{0}^{1}\frac{-\operatorname{li}(x)}{\Gamma(x)}dx<\frac{1}{3\gamma}$

$$\gamma<\int_{0}^{1}\frac{-\operatorname{li}(x)}{\Gamma(x)}dx<\frac{1}{3\gamma}$$ Where $\operatorname{li}(x)$ is the Logarithmic integral function $\Gamma(x)$ is the Gamma function $\gamma$ ...
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The, kind of, truncated moment of the generalized Gamma distribution for positive $a,d$ and $p$ is $$\int_0^\infty (x-k)_+^a x^{d-1}e^{-x^p}dx=\frac1p\int_0^\infty (y^{\frac1p}-k)_+^a\,y^{\frac dp-1}e^... 0answers 21 views ### Comparision between the incomplete gamma function and the gamma function for complex argument Is the lower incomplete gamma function (https://en.wikipedia.org/wiki/Incomplete_gamma_function#Definition) bounded by the gamma function in the right half plane or in a strip parallel to the ... 1answer 16 views ### Limit of ratio of incomplete gamma function In order to derive Sterling's approximation, I need to show that the following integral decays quicker than at least \mathcal{O}(n^2): \lim_{n\to\infty}\dfrac{\int_{2n}^\infty x^ne^{-x}dx}{\int_{0}^... 1answer 35 views ### “generalised” gamma-like integral \int_0^\infty x^ne^{-f(n)x}dx I have noticed that if we have an integral of the form:$$I[f]=\int_0^\infty x^ne^{-f(n)x}dx=\frac{1}{f^{n+1}(n)}\int_0^\infty x^ne^{-x}dx=\frac{n!}{f^{n+1}(n)}$$I was wondering what kind of ... 0answers 12 views ### Integral of upper incomplete gamma function Can anyone please help me with the integral below. I would like to know whether the following relation is correct? \int\big(\frac{t}{A}\big)^{n-1}\exp\big(-\frac{t}{A}\big) dt = -A\Gamma(n, \frac{t}{... 1answer 20 views ### How to calculate the following Laplace transform:  \mathcal{L}[\frac{1-J_0(t)}{t}] ? I'm trying to calculate the Laplace transform of this function.$$ \mathcal{L}[\frac{1-J_0(t)}{t}] $$where J_0(t) is the zeroth Bessel function. Solution Attempt The p-Bessel function is ... 0answers 27 views ### Meaning of Generalized Binomial Coefficients The title speaks for itself. I have seen and understood that a common way to write binomial coefficients in general is$${n \choose k} = \frac{\Gamma{(n + 1)}}{\Gamma{(k + 1)}\Gamma{(n - k + 1)}},$$... 0answers 31 views ### Is this a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{…^{(\frac{1}{n!}-\frac{1}{(n+1)!})}}} have a finit limit? My question here is related to telescopic sum using factorial and it is related to my question here, I have computed some values of a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{...^{(\frac{1}{n!... 1answer 112 views ### What number's factorial is i? I am trying to find the solution to the equation-$$\Gamma(z)=i$$I have tried doing it the following way- LHS is-$$\displaystyle \int_{0}^{\infty}t^ze^{-t}\ dt$$Taking z=a+ib, we get-$$\...
A very famous log-gamma integral due to Raabe is $$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$ Several proofs of this result can be found here. I would like to known about the ...