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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1answer
17 views

Conformity of two concentric annuli

I'm trying to prove that two annuli, $A_1:=\{z:r_1<\left| z \right| < R_1 \}$ and $A_2:=\{z:r_2<\left| z \right| < R_2 \}$ are conformally equivalent if and only if $\frac{R_1}{r_1}=\...
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31 views

Infinite product convergence in complex analysis: $\prod (1+a_k)$ and $\prod|(1+ a_k)|$

If $\prod (1+a_k)$ converges, then to prove that $\prod|(1+ a_k)|$ converges. Any suggestion and idea about this will be highly appreciated. Thank you.
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19 views

Complex limit question [duplicate]

If $z_n\rightarrow z$ as $n\rightarrow \infty$, prove that $\frac{z_1+z_2+...+z_n}{n} \rightarrow z$ as $n\rightarrow \infty$. Could someone please point me in the right direction? I'm not sure where ...
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1answer
16 views

Kind of isolated singularity for analytic function between punctured disc and annulus

If $S_1:=\{z: 0<\left| z\right|<R_1 \}$ and $S_2:=\{z: r<\left| z\right|<R_2 \}$, where $r, R_1, R_2 > 0$, and $\exists f:S_1\to S_2$ such that $f$ is analytic, then what kind of an ...
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1answer
36 views

Proof about triangle inequality

Follow the steps bellow to give an algebraic derivation of the triangle inequality $$|z_1+z_2| \leq |z_1|+|z_2|$$ a) Show that $$|z_1+z_2|^2=(z_1+z_2)(\overline{z_1}+\overline{z_2})=z_1\overline{z_1}...
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1answer
19 views

Proving two punctured domains are conformally equivalent

Prove that $S_1:=\{z:0<\lvert z \rvert<R_1 \}$ and $S_2:=\{z:0<\lvert z \rvert<R_2 \}$ are conformally equivalent. Proof: We need to find an analytic biholomorphic function $f:S_1\to S_2$...
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4answers
63 views

If ${\overline{z}}^2=z^2$ where z is a complex number then z is either real or pure imaginary

If ${\overline{z}}^2=z^2$ where z is a complex number then z is either real or pure imaginary Approach: I approach this algebraically. I set $z=x+yi$ and came up with $(x^2-y^2)-2xyi=(x-y^2)+2xyi$ I ...
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1answer
21 views

Sketch the set of points determined by the following condition

$|2 \overline{z}+i|=4$ if I let $z=x+yi$, I got $4x^2+(1-2y)^2=16$ I don't know if that's right. Should I just plot that in wolfram alpha?
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13 views

Mappings and directional derivatives

In one book on Complex Variables, it is said that if the function $h(u,v) = v+2$, the transformation $w=iz^2=i(x+iy)^2=-2xy+i(x^2-y^2)$ is conformal when $z\ne 0$. It maps $y=x$ (for $x>0$) onto ...
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2answers
50 views

Show that entire function $f:\mathbb{C}\to \mathbb{C}$ so that $|f(\cos z)|\leq A |z|^n$ is constant

Let $f:\mathbb{C}\to \mathbb{C}$ a holomorphic function so that $|f(\cos z)|\leq A |z|^n$ for all $z$ with $|z|>1$ and positive numbers $A$ and $n$. Show that $f$ is constant.
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28 views

Zeros of Eisenstein Series?

I'm wondering if anyone knows how to, or has seen any literature on, analytically handling zeros of Eisenstein series? Or perhaps, asymptotics on the zeros? For example, I'm specifically interested ...
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1answer
27 views

Implicit change of path of contour integral

I want to compute a contour integral say: $\oint_{C_z} \frac{dz}{S(z)-a}$ or $\oint_{C_z} g(S(z))\,dz$. The problem is that I don't have an explicit form for S(z); rather I have the implicit equation $...
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3answers
69 views

Example of analytic function that maps circle to self intersecting curve

Is there an explicit example that an analytic function maps the unit circle to a self intersecting curve? As unit circle is not homogenous to self intersecting curve, I am considering finding an ...
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3answers
69 views

Show that $f$ is identically zero in $\mathbb{C}$

Let $f$ be holomorphic in $\mathbb{C}$. Prove that if $|f(z)| \leq M|z|^{\alpha}$ with $0 <\alpha <1$, then $f$ is identically zero in$\mathbb{C}$ I know that $f$ has a Taylor expansion $f(...
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0answers
20 views

$f(z)$ analytic in the panctured disk $\{|z|<1 \}\setminus\{0\}, \text{Im}(f(z))>0$, then $z=0$ is removable singularity

Let $f(z)$ be analytic in the panctured disk $\{|z|<1 \}\setminus\{0\}$ and let $\text{Im}(f(z))>0$. Prove $z=0$ is removable singularity. I try to show that $f(z)$ is bounded near $z=0$ but ...
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1answer
43 views

Find all solutions of $e^{e^z}=1$ in the complex space.

Find all solutions of $e^{e^z}=1$ in the complex space. Attempt: $e^{e^z}=1$. Assuming $e^z$ is a complex number, I will start off solving $e^z=e^{x+yi}=1$: $e^x(\cos y+i\sin y)=1\Rightarrow \sin y=...
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1answer
16 views

Linear combination of a real-valued function and its inverse is analytic Implies the real-valued function is analytic.

If $u$ is a real-valued function on a disc $\Delta_R$ such that $u^{-1}+iu$ is analytic on $\Delta_R$, then does this imply that $u$ is analytic on $\Delta_R$? I am actually trying to prove some ...
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14 views

Proving a set is connected using the definition of Relatively Open set Of a set in $C$

I came across this definition Let $U\subset S\subset C$. We say that $U$ is relatively open in $S$ if for every $z_0 \in U$, there is $r > 0$ such that $$D(z_0 ;r)\cap S\subset U$$: Now the ...
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1answer
35 views

True or false on two statements(about entire function)

$f (z)$ is an entire function. If $f (x)$ is bounded for all real x, then f is a constant function. If | f (z)| → ∞ as |z| → ∞, then f is a polynomial. Can you tell me how to judge these 2 ...
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3answers
28 views

Find the real and imaginary parts of an equation

Find the real and imaginary parts of $\frac{1}{3z+2}$ So I have expanded it out to get $\frac{1}{3x+3iy+2}$ Thus giving $Re(\frac{1}{3z+2})=\frac{1}{3x+2}$ and $Im(\frac{1}{3z+2})=\frac{1}{3y}$ ...
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2answers
31 views

Bound on derivatives of holomorphic function

Let $f:U\rightarrow \mathbb{C}$ be a holomorphic function on a open and bounded subset $U$ of $\mathbb{C}$. Then suppose $\frac 1 C <|\frac{df}{dz}|<C$ for a certain constant $C>0$. Does this ...
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1answer
42 views

Hadamard-like complex variable substitution

\begin{align} \frac\pi a &= \int_{-\infty}^\infty dxdye^{-a(x^2+y^2)}\\ \tag{1}&= \int_{-\infty}^\infty dxdye^{-a(x+iy)(x-iy)} \end{align} So far so good. Now introduce a complex variable $z$ ...
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2answers
42 views

Analytic Functions

Prove or give a counter-example: If $f_j(j=1,2,...,n)$ is analytic on the domain $D$ such that $\sum_{j=1}^n |f_j(z)|^2$ is constant on $D$. Then each $f_j$ is a constant function. Inputs: We know ...
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1answer
29 views

Show that if $g$ and $h$ are holomorphic in $\mathbb{C}$ and $g\circ h$ is a no constant polynomial then $g$ and $h$ are polynomials.

Let two entire functions(holomorphic in all $\mathbb{C}$) $g$ and $h$ so that the composition $g\circ h$ is a no constant polynomial. Show that $g$ and $h$ are polynomials.
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1answer
48 views

Derivative of a analytic function at its fixed point [duplicate]

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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0answers
48 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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2answers
58 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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0answers
47 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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2answers
80 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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0answers
40 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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0answers
18 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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0answers
17 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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1answer
56 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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0answers
18 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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0answers
26 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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2answers
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
35 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
28 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
26 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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1answer
46 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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2answers
80 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
53 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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30 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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1answer
31 views

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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1answer
22 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
29 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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1answer
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 ...
4
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2answers
46 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 $z=1$ ...
0
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0answers
32 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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1answer
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 ...