Question

Catalan's constant and $\int_{0}^{2 \pi} \int_{0}^{2 \pi} \ln(\cos^{2} \theta + \cos^{2} \phi) ~d \theta~ d \phi$

score 524 · Answer 1 · 2016-04-11 16:23:50

524

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Catalan's constant and $\int_{0}^{2 \pi} \int_{0}^{2 \pi} \ln(\cos^{2} \theta + \cos^{2} \phi) ~d \theta~ d \phi$

Published on 11 Apr 2016 - 16:23

score 361 · Answer 2 · 2026-03-25 05:36:23

361

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How prove this sum$\sum_{k=0}^{\infty}\frac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\frac{\sqrt{2}\pi^2}{16}$

Published on 25 Mar 2026 - 5:36

score 98 · Answer 3 · 2026-03-25 05:36:07

98

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Catalan constant preliminaries

Published on 25 Mar 2026 - 5:36

score 443 · Answer 4 · 2026-03-25 05:36:13

443

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Indefinite integral $\int \arctan^2 x dx$ in terms of the dilogarithm function

Published on 25 Mar 2026 - 5:36

score 123 · Answer 5 · 2026-03-25 05:36:12

123

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Show that $\int_{0}^{1}\ln\left({1+x\over 1-x}\right){\mathrm dx\over 1+x^2}=C$

Published on 25 Mar 2026 - 5:36

score 114 · Answer 6 · 2026-03-25 05:36:23

114

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Show that $(-1)^n\int_{0}^{\pi/4}(\tan{x})^{2n}\ln(\cot{x})\mathrm dx=G+\sum_{k=1}^{n}{(-1)^k\over (2k-1)^2}$

Published on 25 Mar 2026 - 5:36

score 324 · Answer 7 · 2026-03-25 05:36:13

324

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Evaluate $\int_{0}^{\frac{\pi}{2}} \sinh^{-1}(\sin(x))dx$

Published on 25 Mar 2026 - 5:36

score 954 · Answer 8 · 2026-04-16 10:27:03

954

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Evaluate $\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt$

Published on 16 Apr 2026 - 10:27

score 1.2k · Answer 9 · 2026-04-14 16:04:48

1.2k

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A double series yielding Riemann's $\zeta$

Published on 14 Apr 2026 - 16:04

score 943 · Answer 10 · 2026-04-14 22:39:32

943

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Catalan constant is irrational. What is wrong with this proof?

Published on 14 Apr 2026 - 22:39

score 240 · Answer 11 · 2026-04-10 21:10:23

240

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use complex analysis to evalaute series involving the square of coth

Published on 10 Apr 2026 - 21:10

score 224 · Answer 12 · 2026-03-25 05:36:07

224

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integral arising in statistical mechanics

Published on 25 Mar 2026 - 5:36

score 153 · Answer 13 · 2026-03-25 05:36:12

153

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Evaluate $\int_0^1 \frac{\ln{\left(ax^2+2bx+a\right)}}{x^2+1} \; \mathrm{d}x, \;a>b>0$

Published on 25 Mar 2026 - 5:36

score 266 · Answer 14 · 2026-03-25 05:36:07

266

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Why does this proof fail? Catalan Constant's Exact Value.

Published on 25 Mar 2026 - 5:36

score 475 · Answer 15 · 2026-03-25 05:36:23

475

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How to Evaluate $ \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sum_{k=1}^{n}\frac{1}{4k-1} $

Published on 25 Mar 2026 - 5:36

List Question

Catalan's constant and $\int_{0}^{2 \pi} \int_{0}^{2 \pi} \ln(\cos^{2} \theta + \cos^{2} \phi) ~d \theta~ d \phi$

How prove this sum$\sum_{k=0}^{\infty}\frac{(-1)^{\frac{k(k+1)}{2}}}{(2k+1)^2}=\frac{\sqrt{2}\pi^2}{16}$

Catalan constant preliminaries

Indefinite integral $\int \arctan^2 x dx$ in terms of the dilogarithm function

Show that $\int_{0}^{1}\ln\left({1+x\over 1-x}\right){\mathrm dx\over 1+x^2}=C$

Show that $(-1)^n\int_{0}^{\pi/4}(\tan{x})^{2n}\ln(\cot{x})\mathrm dx=G+\sum_{k=1}^{n}{(-1)^k\over (2k-1)^2}$

Evaluate $\int_{0}^{\frac{\pi}{2}} \sinh^{-1}(\sin(x))dx$

Evaluate $\int_{0}^{\frac{\pi}{4}}\ln(\cos t)dt$

A double series yielding Riemann's $\zeta$

Catalan constant is irrational. What is wrong with this proof?

use complex analysis to evalaute series involving the square of coth

integral arising in statistical mechanics

Evaluate $\int_0^1 \frac{\ln{\left(ax^2+2bx+a\right)}}{x^2+1} \; \mathrm{d}x, \;a>b>0$

Why does this proof fail? Catalan Constant's Exact Value.

How to Evaluate $ \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \sum_{k=1}^{n}\frac{1}{4k-1} $

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