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Question on properties of Gamma functions

score 98 · Answer 1 · 2026-04-13 01:37:29

98

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Question on properties of Gamma functions

Published on 13 Apr 2026 - 1:37

score 67 · Answer 2 · 2026-04-15 08:59:27

67

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How to compute $I_{m,n,k}=\int_{0}^{\infty} \frac{dx}{\sinh^n(x) \coth^{2k+n+m-1}(x)}$?

Published on 15 Apr 2026 - 8:59

score 339 · Answer 3 · 2026-04-16 11:49:52

339

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$\sum_{k=0}^{\infty} \Gamma\left(k+\alpha\right) \, \frac{t^k}{k!}=??$

Published on 16 Apr 2026 - 11:49

score 179 · Answer 4 · 2026-04-16 19:56:45

179

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Prove $\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$

Published on 16 Apr 2026 - 19:56

score 47 · Answer 5 · 2026-05-04 01:33:03

47

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How to compute this integral involving associated Legendre function?

Published on 04 May 2026 - 1:33

score 240 · Answer 6 · 2026-04-09 18:54:03

240

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Hermite polynomials for fractional orders and negative arguments

Published on 09 Apr 2026 - 18:54

score 207 · Answer 7 · 2026-04-17 09:26:50

207

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Integrate $\int_{0}^{2\pi}e^{-[a\cos(2\psi)+b\cos(\psi)]}d\psi$ and $\int_{0}^{2\pi}e^{-[a\cos(\phi+\psi)+b\cos(\phi-\psi)]}d\psi$

Published on 17 Apr 2026 - 9:26

score 174 · Answer 8 · 2026-04-17 09:38:40

174

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Closed form of $\int e^{i\csc^2(x)}dx=\int \cos\left(\csc^2(x)\right)dx+i\int \sin\left(\csc^2(x)\right)dx$

Published on 17 Apr 2026 - 9:38

score 80 · Answer 9 · 2026-04-16 14:58:19

80

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Work done by $\vec F=\sin(x^2)\hat x+(3x-y)\hat y$ needs problematic integral?

Published on 16 Apr 2026 - 14:58

score 221 · Answer 10 · 2026-04-13 23:05:37

221

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Evaluation of $\sum\limits_{n=1}^\infty \frac{(-1)^n\text{Ei}(n)}{n!}$

Published on 13 Apr 2026 - 23:05

score 75 · Answer 11 · 2026-04-12 01:10:20

75

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Closed expression for a given continued fraction

Published on 12 Apr 2026 - 1:10

score 160 · Answer 12 · 2026-04-11 15:54:04

160

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Closed expression for a continued fraction

Published on 11 Apr 2026 - 15:54

score 182 · Answer 13 · 2026-04-16 05:52:45

182

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Gaussian integral $\int_{0}^{\infty} dx \, x \, e^{-a^2 x^2} \left( \sqrt{(x - c)^2 + b^2} - \sqrt{(x + c)^2 + b^2} \right)$

Published on 16 Apr 2026 - 5:52

score 329 · Answer 14 · 2026-04-17 09:42:10

329

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Converting $ \int_0^{\infty} \frac{e^{-\varepsilon s} \, (s+s^2)^{\beta}}{\log^{\gamma}((1+s)/s)} \, ds$ to a sum?

Published on 17 Apr 2026 - 9:42

score 105 · Answer 15 · 2026-04-15 09:20:34

105

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Integral with four modified Bessel functions of the second kind of imaginary order

Published on 15 Apr 2026 - 9:20

List Question

Question on properties of Gamma functions

How to compute $I_{m,n,k}=\int_{0}^{\infty} \frac{dx}{\sinh^n(x) \coth^{2k+n+m-1}(x)}$?

$\sum_{k=0}^{\infty} \Gamma\left(k+\alpha\right) \, \frac{t^k}{k!}=??$

Prove $\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$

How to compute this integral involving associated Legendre function?

Hermite polynomials for fractional orders and negative arguments

Integrate $\int_{0}^{2\pi}e^{-[a\cos(2\psi)+b\cos(\psi)]}d\psi$ and $\int_{0}^{2\pi}e^{-[a\cos(\phi+\psi)+b\cos(\phi-\psi)]}d\psi$

Closed form of $\int e^{i\csc^2(x)}dx=\int \cos\left(\csc^2(x)\right)dx+i\int \sin\left(\csc^2(x)\right)dx$

Work done by $\vec F=\sin(x^2)\hat x+(3x-y)\hat y$ needs problematic integral?

Evaluation of $\sum\limits_{n=1}^\infty \frac{(-1)^n\text{Ei}(n)}{n!}$

Closed expression for a given continued fraction

Closed expression for a continued fraction

Gaussian integral $\int_{0}^{\infty} dx \, x \, e^{-a^2 x^2} \left( \sqrt{(x - c)^2 + b^2} - \sqrt{(x + c)^2 + b^2} \right)$

Converting $ \int_0^{\infty} \frac{e^{-\varepsilon s} \, (s+s^2)^{\beta}}{\log^{\gamma}((1+s)/s)} \, ds$ to a sum?

Integral with four modified Bessel functions of the second kind of imaginary order

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