Question

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

score 3.3k · Answer 1 · 2013-05-18 19:45:49

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Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

Published on 18 May 2013 - 19:45

score 588 · Answer 2 · 2013-05-18 22:43:26

588

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Closed form for n-th anti-derivative of $\log x$

Published on 18 May 2013 - 22:43

score 2.5k · Answer 3 · 2013-05-19 21:49:45

2.5k

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Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

Published on 19 May 2013 - 21:49

score 183 · Answer 4 · 2013-05-20 03:46:28

183

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Summation involving subfactorial function

Published on 20 May 2013 - 3:46

score 2.2k · Answer 5 · 2013-05-20 15:13:43

2.2k

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Closed form for $n$th derivative of exponential of $f$

Published on 20 May 2013 - 15:13

score 674 · Answer 6 · 2013-05-20 19:37:45

674

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How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

Published on 20 May 2013 - 19:37

score 263 · Answer 7 · 2013-05-21 01:11:41

263

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Closed form for the series $\sum\limits_{n=0}^\infty \frac{\exp(\cos(n))}{n!}$

Published on 21 May 2013 - 1:11

score 1.4k · Answer 8 · 2013-05-21 01:14:20

1.4k

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How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?

Published on 21 May 2013 - 1:14

score 2.5k · Answer 9 · 2026-03-27 04:14:58

2.5k

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Closed form for Exponential Conditional Expected Value & Variance

Published on 27 Mar 2026 - 4:14

score 650 · Answer 10 · 2026-03-27 01:47:43

650

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Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$

Published on 27 Mar 2026 - 1:47

score 306 · Answer 11 · 2013-05-22 17:22:26

306

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Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$

Published on 22 May 2013 - 17:22

score 675 · Answer 12 · 2026-03-30 01:52:02

675

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How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$

Published on 30 Mar 2026 - 1:52

score 100 · Answer 13 · 2013-05-24 15:48:23

100

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Infinite sum convergence $ \sum_{i\geq 1}\frac{1}{x^i-y^i}$

Published on 24 May 2013 - 15:48

score 412 · Answer 14 · 2013-05-25 08:41:15

412

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How to prove a generalized integral identity

Published on 25 May 2013 - 8:41

score 1.1k · Answer 15 · 2026-03-30 01:48:04

1.1k

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How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$

Published on 30 Mar 2026 - 1:48

List Question

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

Closed form for n-th anti-derivative of $\log x$

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

Summation involving subfactorial function

Closed form for $n$th derivative of exponential of $f$

How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

Closed form for the series $\sum\limits_{n=0}^\infty \frac{\exp(\cos(n))}{n!}$

How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?

Closed form for Exponential Conditional Expected Value & Variance

Closed form for $\int_1^\infty\int_0^1\frac{\mathrm dy\,\mathrm dx}{\sqrt{x^2-1}\sqrt{1-y^2}\sqrt{1-y^2+4\,x^2y^2}}$

Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$

How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$

Infinite sum convergence $ \sum_{i\geq 1}\frac{1}{x^i-y^i}$

How to prove a generalized integral identity

How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$

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