Question

Holomorphic function mapping unit disc to the "pacman" $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]\}$

score 496 · Answer 1 · 2026-04-09 13:38:20

496

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Holomorphic function mapping unit disc to the "pacman" $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]\}$

Published on 09 Apr 2026 - 13:38

score 270 · Answer 2 · 2026-04-06 16:05:49

270

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Evaluate $\int_0^\infty\frac{\sin(\varphi_1x)}x\frac{\sin\varphi_2x}x\cdots\frac{\sin\varphi_nx}x\frac{\sin(ax)}x\cos(a_1x)\cdots\cos(a_mx) \, dx$

Published on 06 Apr 2026 - 16:05

score 2.1k · Answer 3 · 2026-04-10 07:46:47

2.1k

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How to show these complex conjugates of exponential functions

Published on 10 Apr 2026 - 7:46

score 115 · Answer 4 · 2026-04-09 00:09:07

115

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A rigorous proof of $\Re\int_0^{2\pi}\left(\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log\left(\frac{1}{1-ne^{i\theta}}\right)\right)d\theta=2\pi$

Published on 09 Apr 2026 - 0:09

score 134 · Answer 5 · 2026-04-10 23:29:41

134

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Harmonic(?) integral function

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score 900 · Answer 6 · 2026-04-09 00:26:52

900

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Modulus and argument of a holomorphic function.

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score 191 · Answer 7 · 2026-03-25 06:01:24

191

Views

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score 533 · Answer 8 · 2026-04-09 02:23:31

533

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score 71 · Answer 9 · 2026-03-26 02:11:44

71

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score 738 · Answer 10 · 2026-04-09 09:39:26

738

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Published on 09 Apr 2026 - 9:39

score 166 · Answer 11 · 2018-02-06 02:30:38

166

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score 112 · Answer 12 · 2026-04-11 11:56:21

112

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score 128 · Answer 13 · 2026-03-25 01:16:23

128

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score 236 · Answer 14 · 2026-04-10 23:43:39

236

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score 961 · Answer 15 · 2018-02-06 14:14:54

961

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Bijective analytic functions from the open unit disk to itself

Published on 06 Feb 2018 - 14:14

List Question

Holomorphic function mapping unit disc to the "pacman" $U = \{|z|<1,\ \mathrm{Arg}z \notin [-\frac{\pi}{4},\frac{\pi}{4}]\}$

Evaluate $\int_0^\infty\frac{\sin(\varphi_1x)}x\frac{\sin\varphi_2x}x\cdots\frac{\sin\varphi_nx}x\frac{\sin(ax)}x\cos(a_1x)\cdots\cos(a_mx) \, dx$

How to show these complex conjugates of exponential functions

A rigorous proof of $\Re\int_0^{2\pi}\left(\sum_{n=1}^{\infty}\frac{\mu(n)}{n}\log\left(\frac{1}{1-ne^{i\theta}}\right)\right)d\theta=2\pi$

Harmonic(?) integral function

Modulus and argument of a holomorphic function.

A naive conjecture about Taylor series convergence

Laurent Series problems

A finite-codimensional subspace of the space of [square-integrable] holomorphic functions

What is a complex line, equation/function wise?

Example where $F$ is not continuous on the closed unit disk

Integral of Gamma Functions

Why do Fourier transforms or Characteristic Functions converge when Laplace xforms or Moment-Generating Functions don't?

Showing that a function is harmonic

Bijective analytic functions from the open unit disk to itself

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