Question

For $u$ harmonic and $f = u+iv$ holomorphic, show that $f(z) = \frac{1}{\pi i} \oint_{|\zeta|=r} \frac{u(\zeta)}{\zeta - z} d\zeta - \overline{f(0)}$

score 197 · Answer 1 · 2023-07-31 20:30:40

197

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For $u$ harmonic and $f = u+iv$ holomorphic, show that $f(z) = \frac{1}{\pi i} \oint_{|\zeta|=r} \frac{u(\zeta)}{\zeta - z} d\zeta - \overline{f(0)}$

Published on 31 Jul 2023 - 20:30

score 43 · Answer 2 · 2023-09-02 14:58:01

43

Views

How to show this equality from the Cauchy integral formula

Published on 02 Sep 2023 - 14:58

score 55 · Answer 3 · 2023-09-10 06:48:00

55

Views

Suitable Contour to find the value of the integral: $\int_{0}^{\infty} \frac{x}{x^3+1} dx$

Published on 10 Sep 2023 - 6:48

score 54 · Answer 4 · 2023-09-13 14:37:57

54

Views

Calculation of an Integral Using the Cauchy Formula

Published on 13 Sep 2023 - 14:37

score 60 · Answer 5 · 2023-09-15 14:32:51

60

Views

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Published on 15 Sep 2023 - 14:32

score 42 · Answer 6 · 2023-09-16 12:15:30

42

Views

Cauchy Theorem Integration Issue

Published on 16 Sep 2023 - 12:15

score 25 · Answer 7 · 2023-10-08 17:47:25

25

Views

Question about uniform convergence of uniformly bounded sequence of holomorphic functions on compact subsets

Published on 08 Oct 2023 - 17:47

score 103 · Answer 8 · 2023-10-09 13:26:42

103

Views

Estimate derivative of analytic function as Laurent series

Published on 09 Oct 2023 - 13:26

score 209 · Answer 9 · 2023-10-15 05:12:05

209

Views

Can Cauchy's Theorem be used to solve a contour integral with singularities?

Published on 15 Oct 2023 - 5:12

score 47 · Answer 10 · 2023-10-18 02:18:28

47

Views

Bessel function, integrals equality

Published on 18 Oct 2023 - 2:18

score 116 · Answer 11 · 2023-11-06 23:34:09

116

Views

Understanding the role of domain in the question related to Cauchy theorem for complex integral

Published on 06 Nov 2023 - 23:34

score 77 · Answer 12 · 2023-11-08 12:41:47

77

Views

Prove Cauchy's estimate for analytic functions using power series representation

Published on 08 Nov 2023 - 12:41

score 43 · Answer 13 · 2023-11-09 10:44:08

43

Views

Reference: Cauchy estimate on annulus

Published on 09 Nov 2023 - 10:44

score 76 · Answer 14 · 2023-11-20 10:07:29

76

Views

Using Cauchy's Integral Theorem Solve $\int_{C} e^{\frac{1}{z}}$ where $c$ is $|z| = 1$

Published on 20 Nov 2023 - 10:07

score 53 · Answer 15 · 2023-11-30 04:30:28

53

Views

Stronger form of Cauchy's integral theorem

Published on 30 Nov 2023 - 4:30

List Question

For $u$ harmonic and $f = u+iv$ holomorphic, show that $f(z) = \frac{1}{\pi i} \oint_{|\zeta|=r} \frac{u(\zeta)}{\zeta - z} d\zeta - \overline{f(0)}$

How to show this equality from the Cauchy integral formula

Suitable Contour to find the value of the integral: $\int_{0}^{\infty} \frac{x}{x^3+1} dx$

Calculation of an Integral Using the Cauchy Formula

Evaluating an Integral Using Cauchy's Theorem and Root-Finding

Cauchy Theorem Integration Issue

Question about uniform convergence of uniformly bounded sequence of holomorphic functions on compact subsets

Estimate derivative of analytic function as Laurent series

Can Cauchy's Theorem be used to solve a contour integral with singularities?

Bessel function, integrals equality

Understanding the role of domain in the question related to Cauchy theorem for complex integral

Prove Cauchy's estimate for analytic functions using power series representation

Reference: Cauchy estimate on annulus

Using Cauchy's Integral Theorem Solve $\int_{C} e^{\frac{1}{z}}$ where $c$ is $|z| = 1$

Stronger form of Cauchy's integral theorem

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