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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18 views

Order of entire function.

Show that $$1)\ \ \ \ \ \ \ f(z)=\frac{\Gamma^2(1+d)}{\Gamma(1+d+z)\Gamma(1+d-z)}, \ \ d\in\mathbb R$$ $$2)\ \ \ \ \ \ \ f(z)=\frac{\Gamma^2(1+\bar d)}{\Gamma(1+\bar d+z)\Gamma(1+\bar d-z)}, \ \ \bar ...
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3answers
31 views

Complex analysis. Manipulation of conjugates, fractions and modulus.

Let $a,b,c \in \mathbb C $ with $|b|<1$ and $z\neq \bar a$ and $$\left|\frac {z-a}{z-\bar a}\right| \le |b| $$ Show that, $$|z| \le |a| \frac{1+|b|}{1-|b|}$$ This a revision question I'm ...
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0answers
13 views

Moving the branch cut of the complex logarithm

The complex logarithm is defined as $\log z:=\operatorname{Log} |z|+i\arg z$ , with the branch cut on the non-negative real axis. Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ ...
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1answer
47 views

Prove that $\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$ using Cauchy product

need to prove using Cauchy product for series for all $\left|z\right|<1$ that $$\frac{1}{\sqrt{1-z}}=\sum_{n=0}^{\infty}\frac{1}{4^{n}}\binom{2n}{n}z^{n}$$ (with appropriate branch of the root ...
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0answers
9 views

Laplace transform of a majorated function

I have the following problem. I have an analytic function and I want to show that it is majorated by a convenient function. To do that, it is very helpful to solve the transformed equation. I have a ...
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2answers
37 views

How do i prove that $| \arg z|<\pi/2$?

Let $|1-z|<1$. Then how do i prove that $| \arg z| < \pi/2$? This is geometrically trivial, but i dunno how to prove this precisely
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1answer
19 views

Showing distance from $z_0$ to the line parametrized by $z(t)=w_0+te^{i \theta}$ is

Suppose $w_0$ and $z_0$ are in $\mathbb{C}$ and $\theta$ is a fixed angle with $0 \le \theta \le 2 \pi$. Show that the distance from the point $z_0$ to the line parametrized by $z(t)=w_0+te^{i ...
3
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2answers
77 views

Contour method to solve $\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx$

Prove the following using complex analysis $$\tag{1}\int^\infty_0\frac{\ln(1+x)}{1+x^2}\,dx=\frac{\pi}{2}\ln(2)$$ I found this problem in Schaum's outlines of complex variables. I thought that we ...
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3answers
34 views

Computing $\int_{|z|=2} {dz \over z^2 + 1}$

Goal: To compute $$ \int_{|z|=2} {dz \over z^2 + 1} $$ by decomposition of the integrand in partial fractions. Attempt: Let $\gamma$ be the circle around the origin of radius $2$. Let us ...
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0answers
17 views

Showing an iterates of a complex function on the upper half plane converges uniformly on compact sets

The following is an irksome problem that my complex analysis class is having trouble solving: Let $*$ be an operator that takes a function $F:\mathcal{H}\to\mathcal{H}$ to a function ...
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1answer
30 views

Computing ${\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right)$

Let $t \in \mathbb{R}$. Is the following elementary calculation correct? $$ {\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right) = \underbrace{{\mathrm{d} \over \mathrm{d}t}\left(it\right) \cdot ...
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1answer
27 views

Showing the winding number of the unit circle is $1$

Let $\gamma$ denote the unit circle parameterized on the domain $[0,2\pi]$. I'm trying to compute $n(\gamma, 0)$ as follows: $$ n(\gamma,0) = {1 \over 2\pi i}\int_\gamma {dz \over z} = {1 \over 2 ...
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1answer
41 views

Computing $\int_{|z|=1} {e^z \over z}\ dz$

Goal: Let $\gamma$ be the unit circle. Then I aim to compute $$ \int_{|z|=1} {e^z \over z}\ dz = \int_{\gamma} {e^z \over z}\ dz $$ Attempt: Consider that $\gamma$ is a closed curve. Let $a = 0$. ...
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0answers
14 views

Taylor expansion and expansion in powers of z-1

I am trying to expand $z^2/(z+1)^2$ as a Taylor Series. I have acquired its partial fraction decomposition of $z^2/(z+1)^2$ = $(1/6)*(1/(z+1)) + (5/6)(1/(z-5))$. The first term is in the form ...
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0answers
25 views

Level set of a real valued harmonic fucntion

Let $f$ be a real valued harmonic function defined on a neighborhood $U$ of origin in $\mathbb{R}^2$. And $f$ is such that its gradient vanishes at origin. Then how do i show that the set given by ...
2
votes
1answer
15 views

Question regarding pluriharmonic function

A real valued function $f$ defined on an open subset $U$ of $\mathbb{C}^n$ is said to be Pluriharmonic if $$\frac{\partial^2}{\partial z_i\partial\bar{z_j}}f\equiv0,$$ for $1\leq i,j \leq n.$ I was ...
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2answers
27 views

Can anyone explain a residue in fairly simple terms?

I'm studying Complex Analysis and everything up to this point has been pretty straightforward to visualise, but I can't get my head around residues, especially as they seem to have two very different ...
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2answers
34 views

Consistent branch choice

I found in my class notes the following comment regarding branch choice: It is important to choose a branch consistently, otherwise one can get absurd results, for example: $-1 = i^2 = ...
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0answers
18 views

Univalent function and one-to-one function

What is the difference between univalent function and one to one function? I do not know how to be rigorous for this problem. I would appreciate if someone can prove this rigorously?
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0answers
46 views

Saddle point method: a rigorous proof?

I am trying to prove in a fully rigorous way the Saddle Point method for holomorphic functions of 1 complex variable. In books I find only complicated general statements or non-rigorous proofs. Hence ...
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0answers
11 views

Linear Fractional Transforms maps the upper half unit disc onto the first quadrant

Since the LFT(Linear Fractional Transform)preserves the angles, and since $\{|z|=1,\operatorname{Im} z>0\}$ intersects $[-1,1]$ at $-1$ and $1$. So we must map one of the two right angles to the ...
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1answer
32 views

Complex Taylor and Laurent expansions

Let $f(z):=\dfrac{1}{2-z-z^2}, z\in\mathbb{C}\setminus\left\{ {1, -2}\right\}$. i) Express $f$ in the form $\dfrac{A}{1-z}+\dfrac{B}{2+z}$. [Answer to this is $\dfrac{1/3}{1-z}+\dfrac{1/3}{2+z}$]. ...
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1answer
20 views

Example for an entire function of finite order but of infinite type

I'm currently racking my brains for an example as described in the question. I have an example $$e^{e^z}$$ which is of infinite order and infinite type. Question is, does there exist an (entire) ...
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0answers
25 views

Bounded on an union of squares

I would like to do this exercise : Let $\displaystyle h(z) = \pi \mathrm{cotan}(\pi z) = \pi \frac{\cos(\pi z)}{\sin(\pi z)}$. And for $q \in \mathbb{N}^{*}$, let $C_{q}$ be the square in the ...
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1answer
16 views

Finding a reidue at essential singularity $z_0=0$

I'm asked to find the residue at $z_0=0$ of the following functions: A) $f(z)=\frac{sin(z^2)}{z^2}$ B) $f(z)=z^3sin(1/z)$ I find it fairly simple to expand these using a power series but I don't ...
2
votes
1answer
26 views

Inverse of the function $\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$

It can be proved that the function $f:[-1,1]\to \mathbb{C}$ defined by $$f(x)=\frac{(1+x)^2-i(1-x)^2}{(1+x)^2+i(1-x)^2}$$ maps the interval $[-1,1]$ one to one onto the lower part of the unit circle. ...
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2answers
28 views

Showing that ${d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)}$

I'm trying to show that $$ {d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)} $$ However my attempt yields that $$ {d \over dz}\log\left[ z - a \over z - b ...
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0answers
22 views

Deriving that ${d \over dz}\left(\log\ z \right) = {1 \over z}$ in the complex plane

How does one derive that $$ {d \over dz}\left(\log\ z \right) = {1 \over z}\text{?} $$
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0answers
17 views

Derivatives, Cauchy-Riemann Equations [on hold]

Given the function, $w=z^4$ and I want to find the following solutions for this equation, Find real functions u and v such that w=u+iv Show that Cauchy-Riemann equation holds at all points in the ...
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1answer
22 views

Proof in complex variable

I've been trying to do a proof for this and I can't get anywhere: If $f:U\rightarrow\mathbb{C}$ is a complex function, such that for every closed $C^1$ path $\int_\gamma f(z)dz=0$, then $f$ is ...
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1answer
25 views

Understanding why $\int_\gamma {dz \over z - a} = k 2\pi i$ for $\gamma$ a closed curve not passing through $a$

The following is a paraphrased proof from Ahlfors. I bolded the part that is confusing me and asked a question about it at the bottom of this post. Hypothesis: Let $\gamma$ be a closed curve that ...
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votes
2answers
40 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
0
votes
1answer
10 views

What is the integrand of $\int_\gamma d\ \log(z-a)$?

Suppose $\gamma$ is a piecewise differentiable closed curve that does not pass through the point $a \in \mathbb{C}$. I'm reading a proof in Ahlfors that shows under this condition we will obtain $$ ...
0
votes
1answer
29 views

Radius of Convergence of $\sum_{n=0}^{\infty}n^2z^n$

Find the radius of convergence of $$\sum_{n=0}^{\infty}n^2z^n$$ (If it matters, $z$ is a complex variable.) My attempt: The radius of convergence is \begin{align*} ...
2
votes
2answers
32 views

Express each function in the form $u(x,y) + iv (x,y)$

I was doing some homework with complex numbers and I'm stuck with these two, I hope that someone can solve these and clear it up for me. Thank you. ln(1+z) z/(3+z) Samples,
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0answers
16 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
1
vote
1answer
24 views

Finding the locus represented by complex variable equations?

I'm trying to solve these two problems related to complex number but hardly found a solution. I hope that someone can solve these and clear it up for me. Thank you. |z+2|=2|z-1| |z+5|-|z-5|=6
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1answer
12 views

Meromorphic and even

I would like to do the following exercise : Let $f$ be a meromorphic function and $\mathcal{P}$ the set of its poles. We also assume that $f$ is even ($\forall z \in \mathbb{C}, \; ...
4
votes
2answers
36 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
3
votes
2answers
25 views

Analytic $F(z)$ has $f(z)$ as derivative $\implies$ $\int_\gamma f(z)\ dz = 0$ for $\gamma$ a closed curve

Hypothesis: Suppose that $F(z)$ has $f(z)$ as a derivative. Suppose further that $F(z)$ is analytic. Now consider the complex line integral $$ \tag{1} \int_\gamma f(z)\ dz $$ Question: Does this ...
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42 views

Cauchy Integral Theorem problem (lack of understanding)

First of all i was asked to evaluate this integral $\int_\gamma \frac{2z}{(z-1)(z-3)} dz$ where $\gamma (t) = 2e^{it}$ for $0\leq t \leq 2\pi$. Now I thought I would have to calculate this ...
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2answers
73 views

What does $d\bar z$ mean?

What does $d\bar z$ mean? For a manifold, given a local coordinate, $dx$ acts on tangent vectors and gives its corresponding components. What does $d\bar z$ do? The complex field is a one ...
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2answers
29 views

Laplace equation on semidisk

I am interested in the solution of the following boundary value problem on the semidisk $D=\{(r,\theta): 0<r<1, 0<\theta<\pi\}$: $$u_{xx}+u_{yy}=0 \mbox{ in } D, $$ $$u(1,\theta)=0 \mbox{ ...
4
votes
1answer
48 views

Doubt about derivatives in complex variable

Actually, I had this existential doubt while working in my homework. It's obvious that, if I have a sequence $\{z_n\}_{n\in\mathbb{N}}$ that converges to $z_0$, then $$ \dfrac{f(z_n) - f(z_0)}{z_n - ...
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votes
1answer
25 views

Applying Jensen's formula to polynomials?

Prove that $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(e^{i\theta})|^2d\theta=\sum_{k=0}^n|c_k|^2$$ for each polynomial $f(z)=\sum_{k=0}^nc_kz^k$. The hint given by the homework is: show first that for integer ...
1
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1answer
43 views

Laurents Series Expansion Complex Analysis

So here is the problem, I am having a lot of trouble with laurents expansions and if you guys even know any sources where I can learn these really well and very simply then that would be a great help. ...
0
votes
1answer
46 views

Number of zeros of $ z^7+4z^4+z^3+1$

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2? I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number ...
3
votes
0answers
64 views

Independently analytic and continuous, but not jointly continuous?

In Bak/Newman's "Complex Analysis", they write: 17.9 Theorem Suppose $\phi(z,t)$ is a continuous function of $t$, with $b \ge t \ge a$, for fixed $z$ and an analytic function of $z \in D$ for ...
0
votes
0answers
17 views

How to deal with x* when solving complex-variable linear equation(s) of x?

The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form \begin{equation} \sum_i a_i x_i=c\ldots\ldots(1) \end{equation} with $a_i,c\in ...
0
votes
1answer
19 views

Sequence of complex numbers, having throuble with this problem.

The question: Supose $a,b \in \mathbb{C}$ with $\lvert a \rvert = \lvert b\rvert > 1$. If the sequence $\{a^n - b^{n}\}_{n \in \mathbb{N}}$ is limited, prove that a = b. I was thinking in use the ...