The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...

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33 views

Derivative of a analytic function at its fixed point

Let $D$ be a bounded domain, and let $f(z)$ be an analytic function from $D$ to $D$.Show that if $z_{0}$ is fixed point for $f(z)$,then $|f'(z_{0})|\leq 1$ All the conditions above make me think ...
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20 views

A Conformal Mapping problem

Can you give me a proof of this? I know it may use comformal mapping theorem but don't know how to do. Thanks you very much for your help!
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1answer
46 views

Determine the number of roots in complex plane

Prove that the equation $z^{3}e^{z}=1$ has infinitely many complex solutions.How many of them are real? Use the argument principle,I choose a disk centered at $0$ with radius $R$ and get $\int_{\...
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1answer
28 views

Complementary Text to Gunning and Rossi - Analytic functions in several complex variables

I'm currently a second year student who has a background in group theory, ring theory, galois theory, metric spaces and point set topology. I'm currently taking courses in algebraic topology, advanced ...
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73 views

How do I solve equation $\bar{z} = |z|$ correctly?

I'm having troubles, finding how solution would look like for complex equation of the form $\bar{z} = |z|$. Taking $z = x + iy$, we get the following: $$x - iy = \sqrt{x^2 + y^2},$$ then raising it to ...
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27 views

A question about the log of a rational function

We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that : $$\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}=(-1)^{n}\prod_{k=1}^{n-1}\...
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16 views

Symmetry in Analytic Continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we have the following function: $$F(x)=\sum_{n=0}^{\infty} e^{-x E_n}$$ Where $E_n$ is a positive monotonically increasing sequence, bounded from below. Is there a general condition on $E_n$ ...
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15 views

Existence of a holomorphic square root

The problem is as follows. Define\begin{equation*}K=\{iy:y\geq0\}\cup\{x:x\geq0\}\cup\{e^{i\theta}:\frac{3\pi}{4}\leq\theta\leq\frac{7\pi}{4}\}\end{equation*} and $G=\mathbb{C}\setminus K$. Define $f(...
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48 views

Holomorphic function between $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ and $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$

Does there exist a holomorphic function $h$ that sends the set $\{z\in \mathbb{C}: 1\leq |z|\leq 4\}$ to the set $\{z\in \mathbb{C}: 1\leq |z|\leq 2\}$? I tried proving it but I could not. Thanks ...
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21 views

Compute total curvature of a curve

Let $u:[0,2\pi]\rightarrow \mathbb{C}$, $\theta\mapsto e^{i\theta}$, be the unit circle. Let $f:\mathbb{C}^*\rightarrow \mathbb{C}^*$ be a holomorphic function such that $\frac 1 2 <|df|<2$ in ...
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17 views

Analytic continuation of $\sum_{n=0}^{\infty} (E_n)^{-s}$

Suppose $E_n$ is a monotonically increasing sequence. Under what conditions on $E_n$ may the sum $$q(s)=\sum_{n=0}^{\infty} (E_n)^{-s}$$ Be analytically continued from $q(s)$ to $q(-s)$. How would ...
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24 views

A problem about proving some spaces are not conformally equivalent

Consider the unit disk $D$,the complex plane $C$ and extended complex plane $C^{\ast}$.Show that no two of them are conformally equivalence. From Liouville theorem,it's easy to see that disk is not ...
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38 views

A property about a bounded harmonic function

Suppose $f(x,y)$ is a bounded harmonic function function in the unit disk and $f(0,0)=1$. Show that $$\iint_{D}f\left(x,y\right)\left(1-x^{2}-y^{2}\right)dxdy=\dfrac{\pi}{2}$$ I don't understand why ...
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2answers
32 views

Jordan curve in $C^2$

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
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2answers
27 views

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$, suggestion: Reduce this inequality to $(|x|-|y|)^2 \geq0$ (z is a complex number. R stands for real part and Im stands for imaginary part) Approach: Let $...
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2answers
25 views

Question about triangle inequality

By factoring $z^4-4z^2+3$ into two quadratic factors an using the triangle inequality, show that if $z$ lies on the circle $|z|=2$ ($z$ is a complex number) then $$\left|\frac{1}{z^4-4z^2+3}\right| \...
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44 views

Prove that if $z_1*z_2$=0 then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers

Prove that if $z_1*z_2=0$ then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers by using the following property: $|z_1z_2|=|z_1||z_2|$ Approach: if $z_1*z_2=0$ then $$|...
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62 views

Complex Analysis: Zeros of an analytic function

What approach should I take to solve the attached problem. I was looking along the lines of 'Great Picard Theorem', which states that 'If an analytic function f has an essential singularity at a point ...
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1answer
51 views

Residue of $e^{z+\frac1z}$ at $0$.

I am trying to compute $$\text{Res}\: (e^{z+\frac1z}, 0)$$ and can't get a solvable integral using the definition of a residue. I already know other ways to compute residues of poles of arbitrary ...
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28 views

Proof of a necessary and sufficient condition between annuli centered at the origin [duplicate]

What is a simple way to prove that two annuli $A_1 = {z: r_1 < |z| < R_1}$ and $A_2 = {z: r_2 < |z| < R_2}$ are conformally equivalent if and only if $R_1/r_1 = R_2/r_2$, using standard ...
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A subset of holomorphic functions

Let $A:=\{f: f \text{ is holomorphic over }\mathbb{C} \text{, f is not a polynomial, and exist } r_f>0 \text{ (r depends to f) such that } z\in B_r(0) \text{ for all zero of f}\}$. a) if $f\in A$, ...
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21 views

Complement of image of an entire, holomorphic map contains an open disk implies the map is bounded

I need to show the above in order to then use Liouville's Theorem and conclude the map is constant. Thus, I am not sure that the entirety of the map is necessary. $\exists B(x, r) \subseteq f(\mathbb{...
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1answer
26 views

Integral of $\omega\wedge\overline{\omega}$ on Riemann surface

Let $X$ be a Riemann surface of genus $g$ and $\omega$ a meromorphic 1-form on it. I've read that if $\omega$ has just a simple pole in $x\in X$ (and is holomorphic on $X\setminus\{x\}$) then the ...
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20 views

Inversion of lines and circles using explicit parametrizations

Is there a way to parametrize a line and a circle in the complex plane [by $z = z(t)$], to show that under the inversion function $f(z) = 1/z$, a line is mapped either to a line or a circle, and a ...
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44 views

Double periodic entire function

Suppose f is entire and $f(z)=f(z+1)=f(z+\pi)$. Does this imply $f$ is constant? I want to prove that it is constant.I see that it is enough to consider the value of $f(z)$ in between the lines $z=1$ ...
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31 views

A problem about the property of limit of holomorphic function

Suppose $G\subset\mathbb{C}$ is open and connected,let $\left\{ f_{n}:n=1,2\ldots \right\}$ be a uniformly bounded sequence of holomorphic functions on $G$ that convergences uniformly on compact ...
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26 views

Proper maps and their codomains

A continuous map $f:X\to Y$ is called proper map if for every compact $K\subset\subset Y$ the set $f^{-1}(K)$ is compact. Now, if $\mathbb D=\{z\in \mathbb C;|z|<1\}$. Why the map $f:\mathbb D\to ...
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19 views

find a holomorphic function satisfying specific equality

Let $h$ be holomorphic function on a simply connected domain $\Omega$ with no zero in $\Omega$.Show in detail that there exists a holomorphic function $g$ on $\Omega$ where $h\left(z\right)=e^{g\left(...
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1answer
16 views

Calculating the convergence radius of a Taylor expansion

Find the radius of convergence for the Taylor series $$\left(\cot\dfrac{\pi}{100}z\right)=\sum^{\infty}_{n=0}a_{n}\left(z-20\pi\right)^{n}$$ The singularities of this function are the $100n$ where $n$...
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1answer
33 views

Rewriting a complex matrix as a real matrix and changing basis to $(z,\bar z)$

If we have a complex $n\times n$ matrix $A$ and we rewrite it as a real $2n\times 2n$ matrix $B$ by using the identity $z=x+iy$. I don't get how if we change basis from $(x,y)$ to $(z,\bar z)$ that is ...
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1answer
28 views

Difference between branches $[-\pi, \pi)$ and $[\pi, 3\pi)$ of the complex logarithm

I think that both branches just exclude the negative part of the real line. So what's the difference between them then?
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1answer
57 views

For $f$ analytic on $|z|<1$, $|f|\le M$ with $a_1,\ldots,a_n\in \Bbb{D}$ zeros of $f$ show that $|f(0)|\le M \prod |a_j|$

For $f$ analytic in unit disk $\Bbb{D}$ where $|f|\le M$ with $a_1,\ldots,a_n\in \Bbb{D}$ such that $f(a_1)=\cdots=f(a_n)=0$ show that $|f(0)|\le M \prod |a_j|$. I have tried many approaches ...
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15 views

Domain of convergence is complete Reinhardt

Let $W\subset \mathbb C^n$ be a domain such that $W$ is the domain of convergence for a certain power series at the origin. How to show that the interior of $W$ is a complete Reinhardt domain?
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14 views

Minimum vertex cover exact algorithm analysis

An exact algorithm to find a minimum vertex cover in a simple, undirected graph would be based on the following recursive idea: "either a vertex v is in the minimum cover, or all of its neighbors are"....
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24 views

Products of Laurent Series

I'm trying to find the Laurent expansion for $$\frac{e^{1/z^2}}{z - 1}$$ about $z_0 = 0$. Writing the series for $e^{1/z^2}$ and $1/(z-1)$ individually gives $$\frac{e^{1/z^2}}{z - 1} = -\left(\...
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2answers
37 views

Laurent expansions of $\frac{1}{z-1}$

I want to calculate Laurent expansion of $\frac{1}{z-1}$ thtat are valid in the annuli $\begin{align} (a) & \;\;1<|z|<3\\ (b) & \;\;0<|z-3|<2 \end{align}$ For part $(a)$ since $|...
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1answer
26 views

A related Mean Value theorem result for complex functions

Let $f:\mathbb{R}\rightarrow\mathbb{C}$ a differentiable function and I wonder if I can affirm that $$\forall a,b\in \mathbb{R}\,\,\text{we have}\,\,\left|\frac{f(b)-f(a)}{b-a}\right|\leq |f'(c)|\,\,\...
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100 views

Find the value of $\sum_{r=0}^{\left\lfloor\frac{n-1}3\right\rfloor}\binom{n}{3r+1}$

Show that $$\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\ldots=\dfrac{1}{3}\left[ 2^{n-2} + 2\cos{\dfrac{(n-2)\pi}{3}}\right]$$ My solution:- $$(1+x)^n=\binom{n}{0}+\binom{n}{1}x+\binom{n}{2}x^2+\...
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1answer
42 views

Is $\int_{\gamma} \sec ^2z \ \mathrm{d}z=0$?

Let $\gamma = \gamma(0;2)$. Is $$\int_{\gamma} \sec ^2z \ \mathrm{d}z$$ equal to $0$? I'm trying to answer this question using only tools like Cauchy Theorem or the Deformation Theorem since ...
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24 views

Removable singularity of derivatives

If a function f has a removable singularity at $z_0$, is $z_0$ always a removable singularity of the derivative of f?
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50 views

function relating four points on the unit disk

How do you actually find a holomorphic function such that $f(z_1)= w_1$ and $f(z_2) = w_2$?
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39 views

Fourier transform of $1/z$

What is Fourier transform of the complex function $f: \mathbb C \to \mathbb C$ defined by $f(z)=1/z$? I want to know how to interpret the integral. Is it equal to $\hat{f} (\xi)= \int_\mathbb C \frac {...
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33 views

Alternative derivation of Euler's product formula for sine

Euler's product formula states that: $$\sin(x)=x\prod_{n=1}^{\infty}\left[1-\frac{x^2}{\pi^2n^2} \right].$$ There is also a very simple formula for another product representation for the sine ...
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27 views

Fourier transform in complex analysis

What does convolution means in complex analysis? In particular, I want to calculate $\varphi \ast 1/z$, where $\varphi$ is the characteristic function of unit ball in $\mathbb{C}$, i.e. $\varphi= \...
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12 views

Linear functions vs Linearithmic functions complexity

Can we say Linear functions complexity is lower than Linearithmic functions? ie: ...
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1answer
51 views

Power series(e^) in complex analysis [on hold]

How to prove the sum $\sum_{n=1}^{\infty}\frac{e^{2\pi inx}}{n}$ converges for any x $\notin$ $\mathbb{Z}$ ? Thanks.
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1answer
26 views

Analytic function zero in the given disk

I need to show that f(z)=0 for all z \in D(0,2). From the analyticity of f in D(o,2), I know by Cauchy's theorem it's integral in |z|<2 is zeros. And clearly the integrand has a pole at 1/(n+1) ...
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47 views

self-homeomorphism of the circle

$|z|=1$ is the unit circle in the complex plane. Suppose $g$ is a self-homeomorphism of this circle of order $n$, $n \in \mathbb{N}$, and $g$ acts freely. Is that true that $g$ must be defined by $z ...
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35 views

Contour integral of $\sqrt[3]{z^3-1}$ on $|z| = 2$ and branches

This is an exam question that i'm trying to figure out. Apart from the title, there is a note added to the question that says that: The branch of $\sqrt[3]{}$ is the one that has $\sqrt[3]{7} \in \...
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1answer
33 views

Complex Differentiation using definition

The function $f:\mathbb{C}\to \mathbb{C}$, $f(x+iy)=y+ix$, is not $\mathbb{C}$-differentiable in any point, because the Cauchy-Riemann equations do not hold. It's $\frac{\partial u}{\partial y}=1\neq-...