Question

What is contour integration

score 8.7k · Answer 1 · 2026-04-12 12:37:21

8.7k

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What is contour integration

Published on 12 Apr 2026 - 12:37

score 469 · Answer 2 · 2026-04-09 00:43:05

469

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How to know the contour of any function for the Cauchy integral formula

Published on 09 Apr 2026 - 0:43

score 1.6k · Answer 3 · 2026-04-14 11:57:49

1.6k

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Integral of $z^{n} \log z $ on the unit circle under two assumptions

Published on 14 Apr 2026 - 11:57

score 1.1k · Answer 4 · 2013-07-23 21:58:04

1.1k

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Contour Integrals of Non-holomorphic functions

Published on 23 Jul 2013 - 21:58

score 5.4k · Answer 5 · 2013-07-25 00:14:47

5.4k

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show that $\int_{0}^{\infty } \frac {\cos (ax) -\cos (bx)} {x^2}dx=\pi \frac {b-a} {2}$

Published on 25 Jul 2013 - 0:14

score 2.6k · Answer 6 · 2013-07-25 00:31:40

2.6k

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show that $\int_{0}^{\infty } \frac{\sin (ax)}{x(x^2+b^2)^2}dx=\frac{\pi}{2b^4}(1-\frac{e^{-ab}(ab+2)}{2})$

Published on 25 Jul 2013 - 0:31

score 881 · Answer 7 · 2026-04-09 08:33:47

881

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Intuitive reason for why many complex integrals vanish when the path is "blown-up"?

Published on 09 Apr 2026 - 8:33

score 2.9k · Answer 8 · 2013-07-27 01:15:26

2.9k

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show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

Published on 27 Jul 2013 - 1:15

score 136 · Answer 9 · 2013-07-27 01:25:53

136

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show that $\int_{-\infty}^{\infty} \frac {(\sin x) (x^2+a^2)}{x(x^2+b^2)}dx=\frac{\pi(a^2+e^{-b}(b^2-a^2))}{b^2}$

Published on 27 Jul 2013 - 1:25

score 287 · Answer 10 · 2013-07-28 00:40:55

287

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show that $\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$

Published on 28 Jul 2013 - 0:40

score 284 · Answer 11 · 2026-04-14 00:57:38

284

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show that $\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$

Published on 14 Apr 2026 - 0:57

score 2.8k · Answer 12 · 2013-07-29 01:40:37

2.8k

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show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $

Published on 29 Jul 2013 - 1:40

score 1.6k · Answer 13 · 2026-04-13 16:09:18

1.6k

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closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$

Published on 13 Apr 2026 - 16:09

score 753 · Answer 14 · 2026-04-14 12:15:33

753

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Using the estimation lemma

Published on 14 Apr 2026 - 12:15

score 3.1k · Answer 15 · 2013-07-30 14:09:42

3.1k

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contour integration of a function with two branch points: $\int^{\infty}_0 \frac{\log(1+x)}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$

Published on 30 Jul 2013 - 14:09

List Question

What is contour integration

How to know the contour of any function for the Cauchy integral formula

Integral of $z^{n} \log z $ on the unit circle under two assumptions

Contour Integrals of Non-holomorphic functions

show that $\int_{0}^{\infty } \frac {\cos (ax) -\cos (bx)} {x^2}dx=\pi \frac {b-a} {2}$

show that $\int_{0}^{\infty } \frac{\sin (ax)}{x(x^2+b^2)^2}dx=\frac{\pi}{2b^4}(1-\frac{e^{-ab}(ab+2)}{2})$

Intuitive reason for why many complex integrals vanish when the path is "blown-up"?

show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$

show that $\int_{-\infty}^{\infty} \frac {(\sin x) (x^2+a^2)}{x(x^2+b^2)}dx=\frac{\pi(a^2+e^{-b}(b^2-a^2))}{b^2}$

show that $\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$

show that $\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$

show that $\int_{0}^{\infty}\frac{x\cos ax}{\sinh x}dx=\frac{\pi^2}{4} \operatorname{sech}^2 \left(\frac{a\pi}{2}\right) $

closed form of $\int_{0}^{2\pi}\frac{dx}{(a^2\cos^2x+b^2\sin^2x)^n}$

Using the estimation lemma

contour integration of a function with two branch points: $\int^{\infty}_0 \frac{\log(1+x)}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$

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