Question

Prove that a holomorphic function injective in an annulus is injective in the whole ball

score 1.3k · Answer 1 · 2016-06-30 13:38:32

1.3k

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Prove that a holomorphic function injective in an annulus is injective in the whole ball

Published on 30 Jun 2016 - 13:38

score 1k · Answer 2 · 2016-06-30 14:22:15

1k

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Show that $\operatorname{Res}\left(\frac{1}{f}, z_0\right)=\frac{1}{f'(z_0)}$

Published on 30 Jun 2016 - 14:22

score 232 · Answer 3 · 2016-07-02 20:56:38

232

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holomorph vs having a primitive

Published on 02 Jul 2016 - 20:56

score 101 · Answer 4 · 2016-07-04 21:38:00

101

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Elegent argument for the convergence of $\sum_{n = 1}^{\infty} \frac{\cos(n)}{n^z}$?

Published on 04 Jul 2016 - 21:38

score 1.4k · Answer 5 · 2013-05-21 09:58:24

1.4k

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Understanding the (Partial) Converse to Cauchy-Riemann

Published on 21 May 2013 - 9:58

score 742 · Answer 6 · 2014-02-04 10:34:57

742

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A condition implying that a holomorphic function is the identity map

Published on 04 Feb 2014 - 10:34

score 1.4k · Answer 7 · 2014-02-22 14:00:35

1.4k

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If $f$ is holomorphic and $\lvert\, f\rvert$ is constant, then $f$ is constant.

Published on 22 Feb 2014 - 14:00

score 670 · Answer 8 · 2017-04-09 12:11:48

670

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Holomorphic function with Laplace operator

Published on 09 Apr 2017 - 12:11

score 95 · Answer 9 · 2017-04-11 19:42:09

95

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$V:=\mathbb C \setminus \{1,2,...,n\}$ where $n\ge 3$ ; then is $Aut(V)$ is a finite group of order at most $n!$?

Published on 11 Apr 2017 - 19:42

score 335 · Answer 10 · 2026-02-23 08:07:02

335

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$\{f_n\}$ be a normal family on $V$ s.t. $\{z \in V : \lim _{n\to \infty} f_n(z)$ exists $\}$ has a limit point in $V$

Published on 23 Feb 2026 - 8:07

score 330 · Answer 11 · 2026-03-25 09:29:02

330

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$\{a_n\} \subseteq \mathbb C$ discrete set with no limit point . For any sequence $\{z_n\} $ , there is entire $f $ on $\mathbb C$ with $f(a_n)=z_n$?

Published on 25 Mar 2026 - 9:29

score 1.1k · Answer 12 · 2026-03-26 13:51:39

1.1k

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on the (double) discontinuity of dilogarithm along a branch cut

Published on 26 Mar 2026 - 13:51

score 2k · Answer 13 · 2017-04-17 17:25:41

2k

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Prove complex function is not holomorphic using definition (without Cauchy-Riemann equations)

Published on 17 Apr 2017 - 17:25

score 951 · Answer 14 · 2017-04-20 12:43:53

951

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Show that $\exists g$ holomorphic st $e^{g(z)} = f(z)$ with given $f$ holomorphic and $\int_\gamma \frac{f'(z)}{f(z)}dz = 0$

Published on 20 Apr 2017 - 12:43

score 271 · Answer 15 · 2026-03-25 09:29:48

271

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$f_n \to f$ uniformly on compact subsets of $D$ ; $f$ non-constant , then there is a sequence $\{z_n\}$ s.t. $f_n(z_n)=f(z) , \forall n >N$

Published on 25 Mar 2026 - 9:29

List Question

Prove that a holomorphic function injective in an annulus is injective in the whole ball

Show that $\operatorname{Res}\left(\frac{1}{f}, z_0\right)=\frac{1}{f'(z_0)}$

holomorph vs having a primitive

Elegent argument for the convergence of $\sum_{n = 1}^{\infty} \frac{\cos(n)}{n^z}$?

Understanding the (Partial) Converse to Cauchy-Riemann

A condition implying that a holomorphic function is the identity map

If $f$ is holomorphic and $\lvert\, f\rvert$ is constant, then $f$ is constant.

Holomorphic function with Laplace operator

$V:=\mathbb C \setminus \{1,2,...,n\}$ where $n\ge 3$ ; then is $Aut(V)$ is a finite group of order at most $n!$?

$\{f_n\}$ be a normal family on $V$ s.t. $\{z \in V : \lim _{n\to \infty} f_n(z)$ exists $\}$ has a limit point in $V$

$\{a_n\} \subseteq \mathbb C$ discrete set with no limit point . For any sequence $\{z_n\} $ , there is entire $f $ on $\mathbb C$ with $f(a_n)=z_n$?

on the (double) discontinuity of dilogarithm along a branch cut

Prove complex function is not holomorphic using definition (without Cauchy-Riemann equations)

Show that $\exists g$ holomorphic st $e^{g(z)} = f(z)$ with given $f$ holomorphic and $\int_\gamma \frac{f'(z)}{f(z)}dz = 0$

$f_n \to f$ uniformly on compact subsets of $D$ ; $f$ non-constant , then there is a sequence $\{z_n\}$ s.t. $f_n(z_n)=f(z) , \forall n >N$

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