Question

Prove that $f(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}\ d\zeta,$ for all $z\in\mathbb{C}$ with $|z| \gt 2.$

score 753 · Answer 1 · 2020-09-10 07:09:56

753

Views

Prove that $f(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}\ d\zeta,$ for all $z\in\mathbb{C}$ with $|z| \gt 2.$

Published on 10 Sep 2020 - 7:09

score 492 · Answer 2 · 2020-09-15 19:34:32

492

Views

Using Local Cauchy Theorem, find the integrals $\int_{|z|=1} \sqrt{9-z^2}\, dz$ and $\int_{|z|=1}(z^2+2z)^{-1}\, dz$.

Published on 15 Sep 2020 - 19:34

score 248 · Answer 3 · 2020-09-16 11:27:48

248

Views

Cauchy's integral theorem proof from Green's theorem

Published on 16 Sep 2020 - 11:27

score 324 · Answer 4 · 2020-09-19 21:37:33

324

Views

Uniform bound for derivatives of holomorphic function on compact set

Published on 19 Sep 2020 - 21:37

score 92 · Answer 5 · 2020-09-30 23:56:26

92

Views

Is there a discrete version of the residue theorem?

Published on 30 Sep 2020 - 23:56

score 338 · Answer 6 · 2020-10-08 23:38:59

338

Views

Cauchys integral formula with an exponential

Published on 08 Oct 2020 - 23:38

score 77 · Answer 7 · 2020-10-09 16:15:53

77

Views

Proof of integral representation of Nevanlinna functions without conformal mapping

Published on 09 Oct 2020 - 16:15

score 75 · Answer 8 · 2020-10-12 00:59:19

75

Views

Calculate $\oint_\gamma \frac{\ln(\overline{z} - b)}{z - a} dz$ using the Cauchy-Pompeiu formula

Published on 12 Oct 2020 - 0:59

score 66 · Answer 9 · 2020-10-12 01:42:58

66

Views

Cauchys integral formula question

Published on 12 Oct 2020 - 1:42

score 480 · Answer 10 · 2020-10-12 23:01:05

480

Views

Computing $\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz$ using Cauchy integral formula

Published on 12 Oct 2020 - 23:01

score 177 · Answer 11 · 2020-10-14 00:44:37

177

Views

Complex Analysis Gamelin Chapter IV - Section 4 - Exercise 2 [Cauchy integral formula and $C^\infty$ harmonic functions]

Published on 14 Oct 2020 - 0:44

score 60 · Answer 12 · 2020-10-20 19:42:04

60

Views

Need help with Cauchy's integral $\frac{1}{2\pi i} \int_{c_1(1)} \frac{\sin(z)}{z^{n+1}} \mathrm{d}z$ value the integral

Published on 20 Oct 2020 - 19:42

score 79 · Answer 13 · 2020-10-24 18:51:02

79

Views

If $f:B_2 (0) \rightarrow \mathbb{C}$ is holomorphic such that $f(\mathbb{S}^1)\subset B_r(z_0)$ then $f(\mathbb{B}^2)\subset B_r(z_0)$

Published on 24 Oct 2020 - 18:51

score 96 · Answer 14 · 2020-10-25 18:39:26

96

Views

An integral of an holomorphic and bounded function with singularity

Published on 25 Oct 2020 - 18:39

score 43 · Answer 15 · 2020-10-26 16:16:39

43

Views

Minimum $\Re$ on the circle

Published on 26 Oct 2020 - 16:16

List Question

Prove that $f(z)=\frac{1}{2\pi i}\int_{\Gamma}\frac{f(\zeta)}{z-\zeta}\ d\zeta,$ for all $z\in\mathbb{C}$ with $|z| \gt 2.$

Using Local Cauchy Theorem, find the integrals $\int_{|z|=1} \sqrt{9-z^2}\, dz$ and $\int_{|z|=1}(z^2+2z)^{-1}\, dz$.

Cauchy's integral theorem proof from Green's theorem

Uniform bound for derivatives of holomorphic function on compact set

Is there a discrete version of the residue theorem?

Cauchys integral formula with an exponential

Proof of integral representation of Nevanlinna functions without conformal mapping

Calculate $\oint_\gamma \frac{\ln(\overline{z} - b)}{z - a} dz$ using the Cauchy-Pompeiu formula

Cauchys integral formula question

Computing $\int_{\alpha}\frac{1}{(z-a)^n(z-b)^m}dz$ using Cauchy integral formula

Complex Analysis Gamelin Chapter IV - Section 4 - Exercise 2 [Cauchy integral formula and $C^\infty$ harmonic functions]

Need help with Cauchy's integral $\frac{1}{2\pi i} \int_{c_1(1)} \frac{\sin(z)}{z^{n+1}} \mathrm{d}z$ value the integral

If $f:B_2 (0) \rightarrow \mathbb{C}$ is holomorphic such that $f(\mathbb{S}^1)\subset B_r(z_0)$ then $f(\mathbb{B}^2)\subset B_r(z_0)$

An integral of an holomorphic and bounded function with singularity

Minimum $\Re$ on the circle

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