Question

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

score 210 · Answer 1 · 2026-04-02 16:57:05

210

Views

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

Published on 02 Apr 2026 - 16:57

score 169 · Answer 2 · 2026-04-08 19:46:38

169

Views

unobvious cauchy integral formula

Published on 08 Apr 2026 - 19:46

score 660 · Answer 3 · 2015-03-08 18:11:42

660

Views

cauchy int formula, function not holomorphic

Published on 08 Mar 2015 - 18:11

score 272 · Answer 4 · 2015-03-08 21:27:55

272

Views

Various evalutions of $\int_0^\infty \sin x \sin \sqrt{x} \,dx$

Published on 08 Mar 2015 - 21:27

score 282 · Answer 5 · 2026-04-02 16:52:54

282

Views

Complicated contour integral to be solved.

Published on 02 Apr 2026 - 16:52

score 277 · Answer 6 · 2026-04-09 00:16:27

277

Views

Evaluating an integral using Cauchy's Integral Formula

Published on 09 Apr 2026 - 0:16

score 152 · Answer 7 · 2026-04-09 00:25:20

152

Views

Spectral representation of an analytic function

Published on 09 Apr 2026 - 0:25

score 381 · Answer 8 · 2026-03-25 20:34:05

381

Views

Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

Published on 25 Mar 2026 - 20:34

score 100 · Answer 9 · 2015-03-09 17:15:48

100

Views

$PV \int_{ia-\infty}^{ia+\infty}\frac{e^{ikt}}{\sqrt{t^2-b^2}}dt$

Published on 09 Mar 2015 - 17:15

score 108 · Answer 10 · 2026-04-08 22:59:08

108

Views

Simple Complex analysis integration

Published on 08 Apr 2026 - 22:59

score 234 · Answer 11 · 2015-03-10 20:22:45

234

Views

Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$

Published on 10 Mar 2015 - 20:22

score 2.8k · Answer 12 · 2015-03-10 21:13:33

2.8k

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About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$

Published on 10 Mar 2015 - 21:13

score 3.3k · Answer 13 · 2015-03-11 02:43:57

3.3k

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Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis

Published on 11 Mar 2015 - 2:43

score 499 · Answer 14 · 2015-03-11 18:06:47

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$

score 201 · Answer 15 · 2015-03-11 22:25:08

201

Views

Prove that $f=u+iv$ is differentiable if and only if $\lim_{r→0} \frac{1}{πr^2 } \int_{C(z_0,r)}f(z)dz=0$

Published on 11 Mar 2015 - 22:25

List Question

Show that $ \int^1_0 x^3 \sqrt{x} \sqrt{1-x} dx = \frac{\pi}{5!} \frac{1.3.5.7}{2^5} $

unobvious cauchy integral formula

cauchy int formula, function not holomorphic

Various evalutions of $\int_0^\infty \sin x \sin \sqrt{x} \,dx$

Complicated contour integral to be solved.

Evaluating an integral using Cauchy's Integral Formula

Spectral representation of an analytic function

Evaluate Integral with $e^{ut}\ \Gamma (u)^{2}$

$PV \int_{ia-\infty}^{ia+\infty}\frac{e^{ikt}}{\sqrt{t^2-b^2}}dt$

Simple Complex analysis integration

Show that $\lim_{r\to 0} \frac{1}{r^2}\int_{C_{r}}f(z)dz = 2 \pi i\frac{\partial f}{\partial \bar{z}}(z_0)$

About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$

Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis

Evaluate $∫_γ \frac{z^2+1}{z(z^2+4)} dz$ Where $γ(t)=re^{it}$ with $0≤t≤2π$ for all possible value of $r$, $0<r<2$ and $2<r<∞$

Prove that $f=u+iv$ is differentiable if and only if $\lim_{r→0} \frac{1}{πr^2 } \int_{C(z_0,r)}f(z)dz=0$

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