Questions tagged [special-functions]

Questions on special functions, useful functions that frequently appear in pure and applied mathematics (usually not including "elementary" functions).

653 questions
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Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the ...
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Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx .$$ I tried to integration by parts and to get a pattern or to ...
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Calculating alternating Euler sums of odd powers

Definition $$\mathbf{H}_{m}^{(n)}(x) = \sum_{k=1}^\infty \frac{H_k^{(n)}}{k^m} x^k\tag{1}$$ We define $$\mathbf{H}_{m}^{(1)}(x) = \mathbf{H}_{m}(x)=\sum_{k=1}^\infty \frac{H_k}{k^m} x^k \tag{2}$$ ...
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Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. [duplicate]

Can anyone give me a hint to evaluate this integral? $$\int_0^\infty \frac{dx}{1+x^4}$$ I know it will involve the gamma function, but how?
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What does | mean?

I found this symbol on Wolfram|Alpha. Does it mean "or"? $$\large \cos^{-1}(-1)=\mathrm{cd}^{-1}(-1\mid 0)$$
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Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$

How to prove $$\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$$ I havent started, any hints?
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Roots of the incomplete gamma function

Is there any way that one can describe all the roots of the incomplete gamma function $\Gamma(n,z)$, for $n\in \mathbb{N}$, analytically?
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Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx$

Prove the following $$\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx = -3\zeta(5)+\pi^2 \frac{\zeta(3)}{3}$$ where $$\operatorname{Li}^2_2(x) =\left(\int^x_0 \frac{\log(1-t)}{t}\,dt \right)^2$$
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Algorithm to compute Gamma function

The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to ...
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Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$

Hi I am trying to prove$$I:=\int_0^1 \log\left(\,\Gamma\left(x+\alpha\right)\,\right)\,{\rm d}x =\frac{\log\left(2\pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha\,,\qquad \alpha \geq 0.$$ I am ...
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Inverse of $y=xe^x$

I feel like finding the inverse of $y=xe^x$ should have an easy answer but can't find it.
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Is there an Inverse Gamma $\Gamma^{-1} (z)$ function?

Since $\Gamma$ is not one to one over the complex domain, Is it possible to define some principal values ( analogues to Principal Roots for the Root function ) so we can have a $\Gamma^{-1} (z)$ (...
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Why isn't the gamma function defined so that $\Gamma(n) = n!$?

As a physics student, I have occasionally run across the gamma function $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$ when we want to generalize the concept of a factorial. ...
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Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

I discovered the following conjecture numerically, but have not been able to prove it yet: $$_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47.\tag1$$ The equality holds ...
Integral that arises from the derivation of Kummer's Fourier expansion of $\ln{\Gamma(x)}$
I am trying to prove that for $0<x<1$, $$\color{blue}{\ln{\Gamma(x)}=\frac{1}{2}\ln(2\pi)+\sum^\infty_{n=1}\left\{\frac{1}{2n}\cos(2\pi nx)+\frac{\gamma+\ln(2\pi n)}{n\pi}\sin(2\pi nx)\right\}}$$...