Question

$f:\mathbb{C} \to \mathbb{C}$ is analytic and $|f(z)| \leq 1$ for all $z$ with $|z|=1$. When is $f(z)=z$?

score 227 · Answer 1 · 2017-06-08 09:35:53

227

Views

$f:\mathbb{C} \to \mathbb{C}$ is analytic and $|f(z)| \leq 1$ for all $z$ with $|z|=1$. When is $f(z)=z$?

Published on 08 Jun 2017 - 9:35

score 2.5k · Answer 2 · 2017-06-15 17:51:59

2.5k

Views

Zeros of $f(z) = z^5+3z^4+9z^3+10$ in the unit disk

Published on 15 Jun 2017 - 17:51

score 340 · Answer 3 · 2017-06-23 08:01:26

340

Views

Roots of $z^4+iz+1$.

Published on 23 Jun 2017 - 8:01

score 374 · Answer 4 · 2017-07-13 15:59:05

374

Views

Multiplicity of zeros of $4z^{10}-e^{z}$ in unit disk

Published on 13 Jul 2017 - 15:59

score 264 · Answer 5 · 2017-07-24 12:57:31

264

Views

$f : B_1(0) \rightarrow \mathbb{C}$ such that $f$ is injective for $\text{Re}z > 0$. What is order of pole at $0$.

Published on 24 Jul 2017 - 12:57

score 3.9k · Answer 6 · 2017-08-02 20:23:41

3.9k

Views

Show that the equation $\lambda - z - e^{-z} = 0$ has exactly one solution in the half plane.

Published on 02 Aug 2017 - 20:23

score 179 · Answer 7 · 2017-08-03 20:14:34

179

Views

Show that $g(w) = \frac{1}{2\pi i}\int_{\gamma}\frac{zf'(z)}{f(z) - w}dz$ is analytic and discuss the properties of $g$.

Published on 03 Aug 2017 - 20:14

score 1.3k · Answer 8 · 2017-08-20 19:16:39

1.3k

Views

Calculate number of zeros using Rouche theorem

Published on 20 Aug 2017 - 19:16

score 588 · Answer 9 · 2017-08-21 01:56:57

588

Views

Rouche's Theorem Application for $z^4-6z+3$

Published on 21 Aug 2017 - 1:56

score 723 · Answer 10 · 2017-08-22 21:47:43

723

Views

How many roots has the polynomial $z^{2015} + 8z^{12} + 1$ in $\{z \in \mathbb{C}: \Im(z) > 0$ and $\Re(z) > 0\}$?

Published on 22 Aug 2017 - 21:47

score 911 · Answer 11 · 2017-10-17 08:52:03

911

Views

Finding the number of roots of $z^4 - 8z + 10$ in an annulus

Published on 17 Oct 2017 - 8:52

score 105 · Answer 12 · 2017-11-13 20:30:00

105

Views

All zeros of $\sum \frac {z^k}{k!}$ in open ball around origin

Published on 13 Nov 2017 - 20:30

score 315 · Answer 13 · 2017-11-28 05:43:45

315

Views

Assumption on Rouche's theorem

Published on 28 Nov 2017 - 5:43

score 163 · Answer 14 · 2020-08-04 16:58:37

163

Views

prove that all the roots of $\sum_{k=0}^{n}\frac{z^{k}}{k!}$ are in $\{z:\frac{n}{e}<|z|<2n\}$

Published on 04 Aug 2020 - 16:58

score 97 · Answer 15 · 2020-08-04 20:02:22

97

Views

Do $e^z+z$ and $ze^z +1$ have the same number of zeros in the strip -$\pi/2 < \text{Im}\,z< \pi/2$

Published on 04 Aug 2020 - 20:02

List Question

$f:\mathbb{C} \to \mathbb{C}$ is analytic and $|f(z)| \leq 1$ for all $z$ with $|z|=1$. When is $f(z)=z$?

Zeros of $f(z) = z^5+3z^4+9z^3+10$ in the unit disk

Roots of $z^4+iz+1$.

Multiplicity of zeros of $4z^{10}-e^{z}$ in unit disk

$f : B_1(0) \rightarrow \mathbb{C}$ such that $f$ is injective for $\text{Re}z > 0$. What is order of pole at $0$.

Show that the equation $\lambda - z - e^{-z} = 0$ has exactly one solution in the half plane.

Show that $g(w) = \frac{1}{2\pi i}\int_{\gamma}\frac{zf'(z)}{f(z) - w}dz$ is analytic and discuss the properties of $g$.

Calculate number of zeros using Rouche theorem

Rouche's Theorem Application for $z^4-6z+3$

How many roots has the polynomial $z^{2015} + 8z^{12} + 1$ in $\{z \in \mathbb{C}: \Im(z) > 0$ and $\Re(z) > 0\}$?

Finding the number of roots of $z^4 - 8z + 10$ in an annulus

All zeros of $\sum \frac {z^k}{k!}$ in open ball around origin

Assumption on Rouche's theorem

prove that all the roots of $\sum_{k=0}^{n}\frac{z^{k}}{k!}$ are in $\{z:\frac{n}{e}<|z|<2n\}$

Do $e^z+z$ and $ze^z +1$ have the same number of zeros in the strip -$\pi/2 < \text{Im}\,z< \pi/2$

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