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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41 views

Periodic-Like Holomorphic Function

I am reading a paper (this one), and the authors (on page 4) implicitly use a result that I do not know. There may be some inaccuracy; I'm inferring the result from the context of the paper. Let ...
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1answer
45 views

Complex Weierstrass M-test question.

Use the Weierstrass M-test to show $\forall\epsilon>0,\sum_{n=1}^{\infty} a_nn^{-z}$ converges uniformly if $Re(z)>=1+\epsilon$, where $a_{n}$ is bounded. This is what I've done: ...
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6answers
190 views

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$

$f(x)=x^3+ax^2+bx+c$ where $1\ge a\ge b\ge c\ge 0$. If $\lambda$ is any root of the polynomial, show that $|\lambda|\le 1$. My attempt: As the polynomial is a cubic, it must have atleast one real ...
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3answers
50 views

Prove that a analytic function is one-one

Is the following statement true? Suppose, $ f: D\to \mathbb C $ is an analytic function, where $ D $ is the disc of radius 1 around 0(including the circle of radius 1). $ |f(z)|=1\forall |z|=1 $. $ ...
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1answer
35 views

Determining a branch of logarithm

The question I have is that what is the explicit mapping that takes the value $-i \pi/2$ at $-i$ where the mapping is a branch of the logarithm in the slit plane $\mathbb{C}- [0,\infty)$? I'm familiar ...
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2answers
48 views

Prove Converse continuity using the Preimages

I would like to prove that if pre images $f^{-1}(U) \subset D $ of open subsets $U\subset \mathbb{C}$ are open in $D$ implies a function $f:D \to \mathbb{C}, D\subset \mathbb{C}$ is continuous. ...
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2answers
23 views

describe and sketch complex set

$f(D)$ where $D = { z : |z|<1}$ and $f(z) = \frac{z+i}{z-1}$. Am I right on saying this set can be described as a translation by $i$ and dilation by $\frac{1}{z-1}$?
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0answers
35 views

Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
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1answer
127 views

About B. Ya Levin's proof that $|f(x)| \leq M$ implies $|f(x+iy)| \leq Me^{\sigma y}$

This question is about Theorems 1 through 3 on pages 37-38 of B. Ya Levin's Lectures on Entire Functions, available on Google Books. If you can't access the Google Books link there is also a ...
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1answer
51 views

Do I correctly understand the constructions involved in definition of Cartier divisor?

Let $(X,\mathcal O)$ be a ringed space, where $\mathcal O$ is a sheaf of unital commutative integrity domains. Let $\widetilde{\mathcal M}_U$ be the field of fractions of ring $\mathcal O_U$ for any ...
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1answer
63 views

Radius of Convergence of product of power series

Is the following statement true? If $P(z)$ is a power series over $\mathbb C$, then $ P(z) $and $P(z)^n$ have same radius of convergence for any positive integer n.
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48 views

One question from complex analysis [duplicate]

Hi I'm taking a modern geometry in this semester and have a question. We are using a complex analysis text book and one homework question asks to prove that if $A+B+C+D=0$, then these points form a ...
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1answer
120 views

Complex Power series with factorials

Find the radius of convergence of $$\displaystyle\sum_{n=0}^\infty z^{n!}$$ $$\displaystyle\sum_{n=1}^\infty {(-1)^nz^{n(n+1)}}/{n}$$ What is the behavior of the series for $z=1, -1,i$
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30 views

if the imaginary part of an entire function f is bounded, then f is constant. [duplicate]

Can you help me to prove the following statement? I know i need to use Liouville's theorem at some point but i just do not how to write a rigorous and complete prroof. "if the imaginary part of an ...
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0answers
48 views

Has anyone used Complex Analysis in the Spirit of Lipman Bers as their textbook?

I have free access to many Springer books from my library, which includes Complex Analysis In the Spirit of Lipman Bers. From what I've seen, it's a decent book that introduces the subject. ...
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1answer
55 views

Norm of a functional on square integrable harmonic functions

Let H be the Hilbert space of square integrable (real) harmonic functions on the unit disk of the complex plane. I want to find the norm of the linear functional $$h\mapsto h_x(0)$$ Here is my proof ...
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1answer
53 views

If $f'(z)\neq 0, \forall z\in \mathbb{C}$, does it necessarily implies that $f$ is one-to-one?

I know that if $f$ is one to one ,then $f(z)\neq 0$, but is it true in the other direction? Thanks.
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1answer
56 views

Cauchy's integral formula used on circle

If $\gamma$ is a piecewise, smooth, positively oriented simple closed curve in $D$, then Cauchy's formula states that $f(z)=1/2\pi i\int_\gamma {f(a)\over {a-z}}$. My textbook also stated that for ...
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0answers
40 views

Prove that if a subset $Z \subset \mathbb{R}^2$ has zero content, a bounded function is integrable on $Z$

Suppose $Z \subset \mathbb{R}^2$ has zero content. Prove that if $f : \mathbb{R}^2 \rightarrow \mathbb{R}$ is bounded, then $f$ is integrable on $Z$ and $\iint_{Z}f \ dA = 0$I'm not sure how to go ...
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1answer
83 views

Help ! bizzare integral [closed]

How to integrate $$ I_1=\underbrace{\int\frac{x^2}{\sqrt{9x^4+4x^2+1}}dx}_{I_1} $$ and $$I_2=\underbrace{\int\frac{dx}{\sqrt{9x^4+4x^2+1}}}_{I_2}$$
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2answers
59 views

evaluate integral by complex method

Can you guys help me how to evaluate this integral by complex analysis method?
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2answers
93 views

Paradox with function representation

Let assume the function $\eta(E)$ has the following representation: $$\eta(E) = \sqrt{\frac{a}{E}}$$ where $a$ is the known positive constant, and $E \in [-\infty, +\infty]$. I know that $\sqrt{a} = ...
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1answer
115 views

The derivative of harmonic function at origin is a bounded linear functional

The following problem is the 5th problem in the qualifying exam of UCLA (spring 2013). Let $\mathbb{D}=\{(x,y):x^2+y^2<1\}$ and let us define a Hilbert space $$H:=\{u:\mathbb{D} \rightarrow ...
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1answer
40 views

Extending a biholomorphic map between two Riemann surfaces

Consider the following problem: $X$ and $Y$ are two compact Riemann surfaces, $S$ is a finite subset of $Y$ and $f:X\longrightarrow Y$ is a holomorphic map whose set of branch points is $S$. Now ...
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1answer
67 views

Integrating around the upper half of $|z|=R$

In a textbook it says that you can show that $ \displaystyle\int_{-\infty}^{\infty} \frac{\cos(x^{2})+\sin(x^{2})-1}{x^{2}} \ dx = 0$ by considering $ \displaystyle f(z) = \frac{e^{iz^{2}}-1}{z^{2}}$ ...
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1answer
46 views

A non constant complex valued function with zero derivative

I need an example of a non constant complex valued function whose derivative is zero at every point in in the complex plane.
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4answers
140 views

Euler's formula for complex $z$

Given Euler's formula $e^{ix} = \cos x + i \sin x$ for $x \in \mathbb{R}$, how does one extend this definition for complex $z$? I.e for $z \in \mathbb{C}, $ show that $e^{iz} = \cos(z) + i\sin(z)$? I ...
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1answer
53 views

Complex Analysis - finding the pole

How do I find the order of the pole of $$\frac{\sin(x^2)}{(x-7)^9(x^5+x+1)^3}$$ at $x = 7$?
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2answers
64 views

Harmonic function vanishing on a set of positive measure.

I'm preparing for a qualifying exam, and came across a question I couldn't figure out: If $\Omega$ is a region and $h:\Omega\to \mathbb{R}$ is a harmonic function vanishing on a set of positive ...
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1answer
41 views

fourier coefficient of an impulse train

HI: I'm going through the signals and systems Schaum's book and I don't understand something that I hope someone could clear up for me. I will repeat the question and then explain the part where I'm ...
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0answers
33 views

Contour Integral of a function

The contour integral $$\int_{C(π/5, π/4)} cot(5x) dx$$ This is what I did: $$cot (5x) = sin(5x)/cos(5x) = 1/5$$ $$2 \pi i \cdot 1.5 =2/5 \pi i$$ Is this how you do it?
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1answer
31 views

How many numbers such it $\cos{\frac{n\pi}{2}}+i\sin{\frac{n\pi}{2}}$is purely imaginary

let $n\in\{1,2,3,\cdots,100\}$, then such $$\cos{\dfrac{n\pi}{2}}+i\sin{\dfrac{n\pi}{2}}$$ is purely imaginary,then How many numbers $n$ such this condition $A:25$ $B:50$ $C:75$ $D:100$ my idea: ...
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1answer
58 views

proving a limit of a function by definition

Consider $f: \Bbb{C} \to \Bbb{C}$ defined by $$ f(z) = \begin{cases} z^3 + 2z &\text{if } z \ne i \\ 3 + 2i &\text{if } z = i \end{cases} $$ Prove that $$ \lim_{z \to i} f(z) = i $$ using the ...
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1answer
57 views

which of the following is/are true for $f$?

Let $f$ be a meromorphic function on $\mathbb C$ such that $|f(z)|\ge|z|$ at each $z$ where $f$ is holomorphic. Then which of the following is/are true? The hypothesis are contradictory. Such an $f$ ...
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0answers
22 views

Show that the function $z\to \displaystyle\int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2]$

Problem: Show that the function $z\to \displaystyle\int_{\sqrt{5}}^{z} \frac{2w}{w^{2}-4}dw$ is well-defined in $\mathbb{C}\setminus [-2,2].$ My approach to this problem was to prove that ...
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2answers
41 views

Convergence of complex power series question

I need some help to solve this problem and find the domain of convergence of the following power series: $$\displaystyle\sum_{n=0}^\infty(2^n+i^n)(z-2i)^n$$ Thank you!
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2answers
74 views

Find $\int_\gamma \frac{dz}{z^2}$ wihtout explicit calculations

Evaluate the following integral without doing any explicit calculations: $\int_\gamma \frac{dz}{z^2}$ where $\gamma(t) = \cos(t) + 2i\sin(t)$ for $0 \le t \le 2\pi$. This exercise comes along with ...
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0answers
331 views

How do I study Stein & Shakarchi's Complex Analysis

I'm currently self-studying some complex analysis. My background is limited: single- and multivariable calculus, linear algebra, introductory Fourier analysis and matrix theory. Each course, with ...
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0answers
40 views

General method of integration when poles on contour

Is there a general method for calculating a contour integral when you have a pole on the contour? For example, how do I integrate, $\frac{1}{z-1}$ over the unit circle centred at the origin?
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1answer
49 views

the fundamental period

Hi : I'm reading the signals and systems schaum's book and I ran into a question based on one of the exercises. This is the most relevant tag I could find so I apologize if there is a better one. How ...
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0answers
44 views

Imaginary part of Log laplacian

I'm confused about how to calculate $\nabla^2 \log z$, where $z=re^{i\theta}$ is a complex number. My calculations return $$ \nabla^2 \log z = 2\pi\frac{\delta(r)}{r} [\delta(\theta) + i ...
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4answers
143 views

Integrating $ \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $

I'm trying to evaluate $\displaystyle \int \limits_{-\infty}^{\infty} \dfrac{\sin^2(x)}{x^2} \operatorname d\!x $. My first though was to use residue calculus, since we've got the pole of order 2 ...
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1answer
63 views

Example of holomorphic function from unit disc to itself

let $f:\mathbb{D} \to \mathbb{D}$ be analytic function with $f(0)=0$,where $\mathbb{D}$ is the open disc $\{z \in \mathbb{C}:|z|<1 \}$ then $1.|f'(0)|=1$ $2.|f(\frac{1}{2})|\leq \frac{1}{2}$ ...
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1answer
51 views

A question on Complex Integral

Consider a complex holomorphic function $f(x+iy)=F(x,y)+iG(x,y)$ where $F,G$ are real-valued. Then how can we express the complex integral $\int f(z)dz$ in terms of the real integrals $\int Fdx,\int ...
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1answer
75 views

Show that two sets are countable infinite

I have to solve the following problem and I was hoping some of you could give me some hints on how to procede. Show that for all $w\in \mathbb{C}$ the sets $$ \{z \in \mathbb{C}: \sin(z) = w\} ...
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1answer
55 views

Series expansion of a complex function

How do I expand a function $f(z)$ in a particular region? For example, how would I expand $f(z)=(z^2-3z+2)^{-1}$ in the region $0<|z-1|<1.$? I believe this can be done by the binomial theorem. ...
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63 views

analytic function $f$ defined in open unit disk for which $f(1/n)$ is $2^n$

How can I show that there does not exists an analytic function $f$ defined in open unit disk for which $f(1/n)$ is $2^n$ .
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1answer
48 views

Complex Fourier series of $f(\theta) = e^{\theta}$

I have the following Fourier series problem: Let $f(\theta)$ be the periodic function such that $f(\theta) = e^\theta$ for $-\pi<\theta\leq\pi\;$, and let ...
1
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3answers
103 views

Integrating $\int _0^\pi \frac{1}{1+\sin^2(\theta)}$ using Cauchy's formula

I need to evaluate $\displaystyle \int _0^\pi \frac{d\theta}{1+\sin^2(\theta)}$ by using Cauchy's integral formula, and the substitution $z = e^{i\theta}.$ So far, I have that $$d\theta = ...
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0answers
36 views

Can every smooth path be parametrized?

How should I go prove this, namely, if $C\in R^n$ is a smooth path, then $\exists r(t),t\in[a,b]$ which describes C. My textbook on line integrals, multi-variable chain rule, etc takes this as ...