# 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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### Analytic continuation of Euler's reflection formula with the Gamma function

Let $\widetilde\Gamma$ be an analytic continuation of $\Gamma$ on $\mathbb C\setminus(-\mathbb N_0)$. Show that the function $$\widetilde\Gamma(z)\widetilde\Gamma(1-z)-\frac{\pi}{\sin(\pi z)}$$ ...
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### Derivative of Incomplete Gamma Function

For the following incomplete Gamma function: $$Γ(1+d,A-c \ln x)=\int_{A-c\ln x}^{\infty}t^{(1+d)-1}e^{-t}dt$$ I am trying to calculate the derivative of $Γ$ with respect ...
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### What is $\int_0^1{\ln\Gamma(x)\sin(\pi x)}\mathop{\mathrm{d}x}$?

Hello I am stuck with this integral: $$\int_0^1{\ln\Gamma(x)\sin(\pi x)}\mathop{\mathrm{d}x}$$ My questions are: What is the integrand, $\ln(\Gamma(x)\sin(x\pi))$ or $\ln(\Gamma(x))\:\sin(x\pi)$? ...
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### customer service time problem

In a disc shop two employees are working. When we get inside the shop, we see that the two employees are already serving two customers (one customer for each employee), with the service time being a ...
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### How to compute $\int_0^1x^a(1-x)^be^{cx}dx$?

How to compute the integral $I(a,b,c) = \int_0^1x^a(1-x)^be^{cx}dx$ ? I know that, $\int_0^1{x^a(1-x)^b}dx = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)}$. Using this result, I tried integration by ...
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### Two complementary continued fractions that are algebraic numbers

Define the two similar continued fractions, $$x=\cfrac{1}{km\color{blue}+\cfrac{(m-1)(m+1)} {3km\color{blue}+\cfrac{(2m-1)(2m+1)}{5km\color{blue}+\cfrac{(3m-1)(3m+1)}{7km\color{blue}+\ddots}}}}\tag1$$...