# Tagged Questions

Questions on the gamma function $\Gamma(z)$ of Euler extending the usual factorial $n!$ for arbitrary argument, and related functions.

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### Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality $\int_{0}^{\infty}e^{-tx}\,dx=\frac{1}{t}$

I'm learning about measure theory, specifically Lebesgue integral, and need help with the following problem: Show that $\int_{0}^{\infty}x^ne^{-x}\,dx=n!$ by differentiating the equality ...
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### Is $\Gamma (0)^{\Gamma (0)^{-1}}$ defined at negative integers / $0$?

Is $x$ here undefined? $$x=Γ(0)^{(1/Γ(0))}$$ I'm aware that for the gamma function: $$\lim_{n\to^{+}0} (Γ(n)) = +\infty$$ $$\lim_{n\to^{-}0} (Γ(n)) = -\infty$$ But for $x$, the limits seem to be: ...
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### Infinite Sum of Incomplete Gamma (with “z” varying)

I am trying to see whether the following series can be "reduced" or re-written in terms of well-known functions. This seems different from other questions on here in that the summation is going ...
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### How are Gauss sums the analogue of the gamma function for finite fields?

I've seen this statement on the internet in a few places and I don't really get the connection. Would anyone mind fleshing out the details? Thanks.
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### Prove that $\lim_{\ r\ \to \ \infty} \dfrac{r! r^x}{x(x+1)(x+2) \dots (x+r)} = \int_{0}^{\infty} t^{x-1} e^{-t} dt$

From Havil & Dyson, "Gamma: Exploring Euler's Constant", section 6.1 I can't prove the following Euler's theorem : ... on 13 October 1729, Euler had already proposed to Goldbach the ...
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### How to prove that $\frac{\Gamma(3/2)}{3\Gamma(11/6)}=\frac{\sqrt{3}\,\Gamma(1/3)^2}{10\pi\sqrt[3]{2}}$?

My task is to show that: $$\frac{\Gamma(3/2)}{3\Gamma(11/6)}=\frac{\sqrt{3}\,\Gamma(1/3)^2}{10\pi\sqrt[3]{2}}.$$ I'm trying to show this using properties of the gamma function, but it seams every ...
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### Is $\Gamma(\alpha, 0, x)$ log-concave as a function of $x$ (for fixed $\alpha$)?

The incomplete Gamma function is defined as: $$\Gamma(\alpha, 0, x) = \int_0^x t^{\alpha - 1} e^{-t} dt$$ Let $\alpha \ge 1$ be a fixed number. Is $\Gamma(\alpha, 0, x)$ log-concave, as a function ...
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### conjectured general continued fraction for the quotient of gamma functions

Given complex numbers $a=x+iy$, $b=m+in$ and a gamma function $\Gamma(z)$ with $x\gt0$ and $m\gt0$, it is conjectured that the following general continued fraction which is symmetric on $a$ and $b$ is ...
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### How to calculate $\int_{0}^{\infty}\frac{dx}{\sqrt{x^{7}}e^{x}}$?

I am trying to calculate the integral $\int_{0}^{\infty}\frac{dx}{\sqrt{x^{7}}e^{x}}$. I worked out the answer to be $\Gamma[-\frac{5}{2}]$, However when I used mathematica to check my answer it says ...
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### Sum of Gamma functions (product pairs)

This is my first time asking a question on stackexchange. Is there an analytical expression for the following summation of Gamma functions? $\sum_{t=0}^m \Gamma (A + t) \Gamma (B + m -t) = ?$ for ...
### Valid Induction Argument involving Closed Form for $\Gamma(1/2-n)$
Given $\Gamma\left(\frac1{2}\right)=\sqrt{\pi}$, $$\sqrt{\pi}=\Gamma\left(\frac1{2}\right)=\Gamma\left(-\frac1{2}+1\right)=-\frac{1}{2}\Gamma\left(-\frac1{2}\right)$$ and so ...