Question

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

score 342 · Answer 1 · 2014-04-12 23:19:48

342

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LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

Published on 12 Apr 2014 - 23:19

score 139 · Answer 2 · 2014-04-12 23:25:53

139

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LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

Published on 12 Apr 2014 - 23:25

score 144 · Answer 3 · 2026-03-27 23:31:16

144

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LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

Published on 27 Mar 2026 - 23:31

score 544 · Answer 4 · 2026-04-02 07:55:49

544

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LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)\mathrm d\theta$

Published on 02 Apr 2026 - 7:55

score 409 · Answer 5 · 2026-03-30 00:22:46

409

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$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

Published on 30 Mar 2026 - 0:22

score 407 · Answer 6 · 2014-04-13 14:20:41

407

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LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Published on 13 Apr 2014 - 14:20

score 292 · Answer 7 · 2026-03-28 10:16:49

292

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Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

Published on 28 Mar 2026 - 10:16

score 110 · Answer 8 · 2014-04-15 14:37:00

110

Views

Prove that $\,f(z) \equiv 0,\,$ if $\,f(z)\,$ is entire

Published on 15 Apr 2014 - 14:37

score 175 · Answer 9 · 2026-03-28 06:48:10

175

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How to handle the complex integration of this function around a branch point

Published on 28 Mar 2026 - 6:48

score 559 · Answer 10 · 2014-04-15 22:45:19

559

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Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Published on 15 Apr 2014 - 22:45

score 263 · Answer 11 · 2014-04-16 17:10:25

263

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Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Published on 16 Apr 2014 - 17:10

score 501 · Answer 12 · 2014-04-16 19:45:27

501

Views

Integral $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Published on 16 Apr 2014 - 19:45

score 59 · Answer 13 · 2014-04-17 18:25:13

59

Views

What is the integrand of $\int_\gamma d\ \log(z-a)$?

Published on 17 Apr 2014 - 18:25

score 1.3k · Answer 14 · 2014-04-18 18:08:37

1.3k

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Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

Published on 18 Apr 2014 - 18:08

score 306 · Answer 15 · 2014-04-18 23:48:15

306

Views

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

Published on 18 Apr 2014 - 23:48

List Question

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

LogSine Integral $\int_0^{\pi/3}\ln^n\big(2\sin\frac{\theta}{2}\big)\mathrm d\theta$

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

Prove that $\,f(z) \equiv 0,\,$ if $\,f(z)\,$ is entire

How to handle the complex integration of this function around a branch point

Integral $\frac{1}{\pi}\int_0^{\pi/3}\log\big( \mu(\theta)+\sqrt{\mu^2(\theta)-1} \big)\ d\theta, \quad \mu(\theta)=\frac{1+2\cos\theta}{2}.$

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Integral $I=\int_0^1\frac{\ln x}{x^n-1}dx$

What is the integrand of $\int_\gamma d\ \log(z-a)$?

Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

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