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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3
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1answer
89 views

Subtlety regarding Rouche's theorem

Given a complex polynomial $f(z)=z^7+z^5-z^4-6z^3-z^2+1$, find the number of roots in $|z|<1$. Now it's clear that for $g(z) = 6z^3$ one can use Rouche's theorem. However, in order to apply it we ...
2
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1answer
55 views

prove that a function is expressible as a power series

I started by rearrange f(z), and expanded the terms in summation. Then, I did not get very far. It would be great if anyone can let me know what is needed to figure out bn. Thanks in advance.
0
votes
1answer
56 views

The sufficient condition for the existence of poles of algebraic function?

Given $P(x,y)=0$, where $P(x,y)$ is a polynomial,and we get the function $y=f(x),P(x,y)=0$, then what is the sufficient condition for the existence of poles of algebraic function $f(x)$? what is the ...
1
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5answers
91 views

Evaluating a contour integral where C is a square

I've been working problems all day so maybe I'm just confusing myself but in oder to do this, I have to the take the integral along each contour $C_1-C_4$ My issue is how to convert to parametric ...
2
votes
1answer
80 views

Evaluate the integral of the given contour

I'm being asked to evaluate $\int \frac{1}{z^3(z^2+1)}dz$, where C is the circle $\lvert z-1 \rvert=\frac32$ I started by determining the zeroes, which are $0, -i, \,i$ Then I applied the Cauchy ...
1
vote
1answer
61 views

What is the Laurent series of $\exp(\frac{1}{z})\exp (2z)$?

What is the Laurent series of $\exp(\frac{1}{z})\exp (2z)$ ? I know how to do the Laurent series of $ \dfrac{1}{z(z+5)}$ (I make use of the geometric series of $\dfrac{1}{1-z}$) but I don't know how ...
0
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0answers
68 views

Excercise for Explicit Riemann Mapping

Set $U:=\{ z\in \mathbb{C}\mid \arg(z)\in(-\alpha,\alpha)\}$ with $\alpha\in(0,\pi)$.Give explicitly the conformal mapping $f:U\rightarrow \mathbb{D}$ such that $f(U)=\mathbb{D}$ with $f(1)=0$ and ...
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1answer
28 views

Existence of $\mathbb{C}$-differentiable function in neighbourhood of $0$

I have a problem with the following: Is there any $\mathbb{C}$-differentiable function in neighbourhood of $0$ such that a) $f(1/n)=(-1)^n \frac{1}{n}$ for $n=1,2,\dots$ b) $f(1/n)=\frac{1}{n^2-1}$ ...
2
votes
2answers
32 views

Simplification for complex numbers

Hi guys I was wondering if it is possible to somehow convert the form $\operatorname{cis}(\theta)$ to the form $(-1)^x$. Is there an algebraic method for this? If so what is it called? Thanks so ...
0
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1answer
31 views

Complex definite integral $\int_{0}^{\pi}\frac{ire^{it}}{2-2ire^{it}}dt$

I am trying to evaluate the integral $$\int\limits_{0}^{\pi}\dfrac{ire^{it}}{2-2ire^{it}}dt$$ but I don't know how to proceed.
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1answer
60 views

Evaluating contour integrals along given C's

Ok, so I have the following problem that I am working on. It says to evaluate $$\int \frac{z}{(z-1)(z-2)}dz$$ where C are given by \begin{align} a)& \ \ C:\lvert z \rvert=\frac12\\ b)& \ \ ...
0
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1answer
54 views

A holomorphic function whose image is contained in the unit circle is constant

I'm looking for a way to proof this without using that holomorphic functions are open mappings. Is there a very basic approach? I'd appreciate any hint :)
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0answers
37 views

asymptotic of a power series

Show that \begin{equation*} \sum_{n=0}^{+\infty}x^{n^{2}} \end{equation*} is equivalent to \begin{equation*} \frac{1}{2}\sqrt{\frac{\pi}{1-x}} \end{equation*} as $x\in (0,1)$ approaches $1$. This ...
0
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1answer
78 views

Uniqueness of Laurent series expansion of $f(z) = \frac{1}{z-1} + \frac{1}{1-z}$.

We first have $$ \frac{1}{z-1} = \frac{1}{z}\frac{1}{1-1/z} = \sum_{n=-\infty}^{-1} z^n.$$ We also have $$\frac{1}{1-z} = \sum_{n=0}^\infty z^n.$$ Now here is the perceived issue. Since $f(z)$ is ...
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0answers
36 views

Does uniform convergence of analytic functions imply pointwise convergence in a bigger domain?

I have a question that seems elementary but I can't figure out the answer. Let $U\subset V\subset \mathbb{C}$ be open and connected. Then let $f_n,f$ be functions analytic and uniformly bounded in V. ...
1
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2answers
45 views

Show that if $f$ is complex differentiable and in some region $U$ it is $f'=0$, then $f$ is constant

I have to show show that if $f$ is complex differentiable and in some region $U$ it is $f'=0$, then $f$ is constant. How can one prove it?
2
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1answer
55 views

How can one justify that mixed second order partial derivatives of $u$ and $v$ where $f$ is a holomorfic function $f(x+iy)=u(x,y)+iv(x,y)$ are equal?

How can one justify that mixed second order partial derivatives of $u$ and $v$ where $f$ is a holomorfic function $f(x+iy)=u(x,y)+iv(x,y)$ are equal? I'm wondering about it but can't find anything.
0
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1answer
32 views

Properties of Harmonic functions and Log

Could anyone advise me on how to prove: If $g$ and $\text{log}|g|$ are harmonic in a simply connected domain $\Omega$, then $g \equiv$ constant on $\Omega.$ Hints will suffice, thank you very much.
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2answers
52 views

Does there exist a $\mathbb{C}$- differentiable function in neighbourhood of $0$ such that $f(1/n)=(-1)^n \cdot \frac{1}{n}$, $n=1,2,\dots$?

Does there exist a $\mathbb{C}$- differentiable function in neighbourhood of $0$ such that $f(1/n)=(-1)^n \cdot \frac{1}{n}$, $n=1,2,\dots$ My attempt: Since we want $f$ to be differentiable at $0$, ...
3
votes
2answers
91 views

If $e^{f(z)}$ is constant can we say that $f(z)$ is constant?

A quick question. If $e^{f(z)}$ is constant can we say that $f(z)$ is constant? ($z \in \mathbb{C}$). Can this be said directly without giving any proof or maybe stating a theorem? Thanks
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0answers
45 views

Use of homological Cauchy theorem

Consider the oriented circumferences $\gamma_1$, $\gamma_2$,$\gamma_3$ such that $\gamma_1$ is positively oriented, $\gamma_2$ and $\gamma_3$ are negatively oriented and contained in the interior of ...
2
votes
1answer
35 views

How to prove coefficients of a power series is bounded?

Let $f(z)=\sqrt{1-z}$. Let $$\sum_{k=0}^\infty c_kz^k$$ be the power series converges to $f(z)$ in the ball $|z|<1$. How can I prove that $|c_k|$ is bounded.
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0answers
49 views

If f is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ then $|f'(0)|\leq 4$

I need to prove that if $f$ is analytic in the unit disk and $|f(z)|\leq \frac{1}{1-|z|}$ for z all $z\in D_1(0)$ then $|f'(0)|\leq 4$. This is my proof and I need to verify this. Let $n\in ...
0
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0answers
34 views

An iterative sequence of complex numbers

Consider a disk at center at $(0,0)$ of radius, $r$ $B_r(0)$ in the complex plane. Let $w_1$ and $w_2$ be two complex numbers belong to the disk $B_r(0)$. Consider a scheme, ...
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1answer
74 views

Homework: Complex integral equation

Suppose $f(z)$ is analytic in the open region $D$, and $C$ is a simple closed curve in $D$. For any $z_0\in D\setminus C$, prove: ...
0
votes
1answer
30 views

Calculate a sum $\sum_{k=0}^{\infty}\cos\frac{k\pi}{6}$

I have to calculate a sum $\sum_{k=0}^{\infty}\cos\frac{k\pi}{6}$. Our lecturer told us we should use de Moivre formula. But i think this sum deosnt even converge...
1
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1answer
32 views

Show that $\Delta u$, $\Delta v= 0$

Let $f=u+iv$ be a complex differentiable function. Then $u$ and $v$ are harmonic. My solution: We have Cauchy-Riemann equations held: $u_x=v_y$ and $u_y=-v_x$. Now: $u_{xx}=v_{yx}$ and ...
1
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1answer
70 views

What is wrong in this calculation $\int_{-\infty}^{\infty}\frac{\cos x}{1+x^2}dx$?

$\def\Res{\operatorname{Res}}\def\Re{\operatorname{Re}}$ First solution: Complex function $f(z)=\frac{\cos z}{1+z^2}$ has a pole $z=i$ on the upper complex plane. $\Res (f,i)=\frac{e+1/e}{4i}$, so $$ ...
1
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2answers
59 views

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

I tried finding the integral of $\int_{|z|=2}^{}\frac{1}{z^2+1}dz$ but not sure whether it is correct. $\gamma(t)=2e^{it},t\in[0,2\pi]$ ...
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2answers
41 views

$\int_{\gamma}^{}\frac{1}{z}dz$, $\gamma$ is the ellipse $x^2+4y^2=1$ traversed once with the positive orientation

I am unable to find the integral $\int_{\gamma}^{}\frac{1}{z}dz$, $\gamma$ is the ellipse $x^2+4y^2=1$ traversed once with the positive orientation. This maybe possible to be done using Cauchy-Goursat ...
0
votes
1answer
63 views

Find the value of the integral on the contour C

Ok, so I'm trying to figure out this problem. It asks to find the value of the contour integral $\dfrac{e^z}{z^2(z-\pi i)}$ on the contour $C$ shown in the following figure I believe that in order ...
1
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1answer
145 views

Rudin's Proof on Riesz Representation Theorem

The proof is from Rudin's Real and Complex Analysis. I am having a hard time understanding part of the Riesz Representation Theorem The Theorem states: Every open set $E$ satisfies ...
5
votes
0answers
66 views

Show that the integral of Riemann function is analytic

I'm trying to resolve this problem. Let $\Omega$ be an open set no empty of $\mathbb C$, $[a,b]$ a compact interval of $\mathbb{R}$, further $f,\ g\colon[a,b] \to \mathbb C$ two integrable Riemann ...
0
votes
1answer
46 views

Integral of a complex function over semicircle enclosing a pole

I have a function with a pole $x_0$ on the real axis. Why would the integral of that function over the countour that a semicircle that is the upper half of the circle enclosing $x_0$. I cannot figure ...
0
votes
1answer
121 views

Arzela-Ascoli equivalent theorems

The following theorems are equivalent? Is the Theorem 2 false? Theorem 1 (Arzela-Ascoli): Let $X$ be a compact metric space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb ...
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1answer
46 views

maps that preserve harmonic functions

Is there a theory of the type of maps between domains that preserve harmonic functions? For instance, in the 2-dimensional case, we know that conformal maps (or even just holomorphic ones) are such ...
2
votes
0answers
236 views

Harmonic functions in the upper half plane

It is a cautionary remark that is often made that solutions to the Dirichlet problem (with continuous boundary conditions) are not unique when the domain in question is the upper half plane. Yes, you ...
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0answers
36 views

Existence of a sequence of polynomials that approximate a holomorphic function uniformly on the closed unit disk [duplicate]

Let $f:D(0,1) \to \mathbb C$ be continuous on the closed unit disc $D(0,1)$ and holomorphic on the open unit disc. Show that there exist a sequence of polynomial that converge uniformly in the closed ...
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3answers
88 views

Stitching two analytic functions?

Let $f$ be an analytic function on the open unit disc and let $g$ be an analytic function on the complement of its closure. Further assume that the two functions have a the same continuous limit on ...
0
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1answer
28 views

Finding the value of a contour integral comprised of line segments

I am attempting to work the following problem but I think am just forgetting a few things in order to answer it. The question asks to find the value of the integral of $(2z+1)dz$ on the contour $C$, ...
3
votes
1answer
78 views

Find complex power series expansion for $\int e^{-w^2} dw$

If a function $E(z)$ is defined on $\mathbb{C}$ by $$ E(z) = \int_0^z e^{-w^2} dw,$$ find a power series expansion for $E(z)$ about $0$. What does this power series converge? I know how this ...
0
votes
1answer
137 views

Equicontinuity and pointwise bounded implies locally uniformly bounded

Is the following proposition true? Let $\mathcal{F}\subset H(\Omega,\mathbb{C})$ be a family of holomorphic functions such that $\bullet$ $\mathcal{F}$ is equicontinuous $\bullet$ ...
2
votes
1answer
36 views

The Laurent series of $f(z)=\frac{1}{e^z-1}$

I am trying to expand $f(z)=\frac{1}{e^z-1}$ in Laurent series. One approach I tried involved writing $f(z)=e^{-z} \frac{1}{1-e^{-z}}$, expanding the fraction as a geometric series: ...
2
votes
1answer
115 views

Proof that $p:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}\setminus\{0\}$ is a covering map, with $p(z)=z^2$

Prove that $p:\mathbb{C}\setminus\{0\}\rightarrow\mathbb{C}\setminus\{0\}$ is a covering map, with $p(z)=z^2$ Let $X=\mathbb{C}\setminus\{0\}$ Let $b\in X$ write $b=re^{j\theta}$ (with ...
0
votes
2answers
111 views

Looking for intuïtive explanation why contour integral of $\frac{dz}{z} $equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this ...
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1answer
73 views

How to integrate $\int_{-1}^1\frac{1}{a + bx }dx$, where $a,b\in \mathbb{C}$ without using branch cuts.

Is there a way to integrate $$\int_{-1}^1\frac{1}{a + bx }dx,\,\,\,\,(*) $$ where $a,b\in \mathbb{C}$ without using branch-cuts? I was approached with such an integral relatively early in my text, and ...
11
votes
2answers
259 views

Calculating the abscissa of convergence for general Dirichlet Series

I'm currently interested in proving this theorem which I have been thinking for quite a while: Define a Dirichlet Series $$\sum_{k=1}^{\infty}a_k e^{-\lambda_k z}$$ where $\lambda_k$ is a strictly ...
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0answers
59 views

Conditions on an analytic function to make it constant

$f:\mathbb C\rightarrow C$ is an analytic function.The question gives 3 conditions and asks to conclude whether it is possible to conclude that $ f$ is constant from here. a.Im$(f^{'}(z))>0$ $ ...
0
votes
2answers
47 views

solution of $\bar{z}=\xi z$

Does $\bar{z}=\xi z$ has solution in $z$ (complex number) for all values of $\xi \in \mathbb{C}$? I was trying to use normal method but its coming quite complicated.
1
vote
1answer
18 views

Why is $f^{(n)}(z)$ real when z is real for all $n$, if this is true for $n=0$?

In order to prove the Schwarz Reflection Principle using the Taylor expansion, we need the information that $f^{(n)}(z)$ is real if $z$ is real for all $n$. We have $f(z)$ real if $z$ is real, and ...