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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229 views

Phragmen-Lindelöf theorem, question from Conway, chapter VI

Page 141, Question 3: Let $G=\{z:|\operatorname{Im} z| < \pi/2\}$ and suppose $f:G\rightarrow C$ and $\limsup|f(z)| \leq M$ on $w$ in the boundary of $G$. Also, suppose $A < \infty$ and $a ...
2
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1answer
68 views

A function in the intersectino of Hardy spaces for $p < 1$

I'm working on an exercise from Peter Duren's Theory of $H^p$ Spaces. The question is: Show that $(1-z)^{-1}$ is in $H^p$ for every $p < 1$, but not in $H^1$. I have been able to show the ...
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0answers
53 views

$U \subset \mathbb{C}$ open, how do I construct $f$ holomorphic, such that there is no continuation of $f$ on any neighbourhood of $z \in \partial U$?

Let $U \subseteq \mathbb{C}$ be open. I want to construct a holomorphic function $f: U \to \mathbb{C}$, such that for all $z \in \partial U$ and for all $\varepsilon > 0$, there is no holomorphic ...
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1answer
50 views

Find condition for positivity of complex-valued functions

The question is as follows. the rational function defined on complex plane $ \displaystyle R(z) = c \cdot \prod_{i=1}^{n} \frac{(z- \alpha_i)(1-\bar \alpha_i z)}{(z-\beta_i)(1-\bar \beta_i z)} $ ...
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1answer
87 views

question about solution of equation complex variable

A friend just told me that the equation $e^{z^2}=0$ has solution. Is it true? Thanks, Dan
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2answers
94 views

Is it possible to calculate this integral using complex analysis?

Evaluate $$ \int_0^\infty\frac{dx}{x^2-2x+4}. $$ I cannot figure it out. Any hint is appreciated.
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1answer
211 views

Extension of Liouville's Theorem?

Liouville's Theorem states that if a function is bounded and holomorphic on the complex plane (i.e. bounded and entire), then it is a constant function. What if we consider the following, slightly ...
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2answers
154 views

Evaluate $ \ \int_{- \infty}^{\infty} \frac{x e^{2ix}}{x^2 - 1}\,dx \ $ using given contour

The question is: Evaluate $\displaystyle \ \int_{- \infty}^{\infty} \frac{x e^{2ix}}{x^2 - 1}\,dx \ $ using the contour below. (Explain what happens on each part of the contour.) First of all, ...
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1answer
258 views

integration of complex trigonometric function

Compute $$\oint_{|z-\frac{\pi}{2}|=\pi+1}z\cdot \tan(z)dz$$ My solution: the integrand is a meromorphic function with simple poles at points: $\frac{\pi}{2}+n\pi$, with $n$ integer. Among these ...
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3answers
64 views

Laurent expansion of $\frac{e^{iz}}{z\,(z^2+1)^2}$ at $z=i$.

I would like to find the residue of $$f(z)=\frac{e^{iz}}{z\,(z^2+1)^2}$$ at $z=i$. One way to do it is simply to take the derivative of $\frac{e^{iz}}{z\,(z^2+1)^2}$. Another is to find the Laurent ...
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55 views

Lower-bounding the distance between zeros of a continuous function

Consider a continuous function of the form: $L(v) = \sum_{i = 0}^{m}[vA_{i} - B_{i}]p^{i}$ where $p$ is the root of the polynomial equation: $vf(p) - g(p) = 0$ with $f(p)$ and $g(p)$ being two ...
5
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1answer
228 views

Showing that infinite product $\prod{(1+\frac{i}{k})}$ diverges

In Bak and Newman's Complex Analysis they ask to show that the infinite product $\prod_{k \ge 1}{(1+\frac{i}{k})}$ diverges (with $i$ being the imaginary unit). My intuition is that it does not ...
2
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3answers
147 views

Evaluate $ \ \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f' (z)}{f(z)}\,dz ,~~\mbox{ where }~ f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} . \ $

How do I evaluate $ \ \frac{1}{2 \pi i} \oint_{|z|=3} \frac{f' (z)}{f(z)}\,dz ,~~\mbox{ where }~ f(z) = \frac{z^2 (z-i )^3 e^z}{3 (z+2 )^4 (3z - 18 )^5} \ $ ? The singularities are z = -2 and z = ...
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1answer
216 views

Questions regarding finite Blaschke product

This problem is totally out of my ability. Not even sure what it is talking about. Somebody please help me to solve this... A finite Blaschke product of degree $n-1$ is a function of the form $ \ ...
3
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3answers
2k views

Some way to integrate $\sin(x^2)$?

Because the straight forward approach involves Fresnel integrals I thought about a different approach of taking the imaginary part of $\int_{-\infty}^{\infty}\exp(ix^2) $ but have no idea how to ...
0
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2answers
62 views

complex analysis

evaluate the principal value of $i^{i+1}$ and derive a general expression for $\arccos(A)$, where $A$ is a real number $> 1$ and hence find $\arccos (3)$, writing your answer in the same form, ...
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1answer
273 views

Books on complex analysis

Is there any book on $1$-dimensional complex analysis, where all is written in the language of sheaf theory? It's clear, that a lot of constructions can be formulated in simplier way using it. There ...
3
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1answer
192 views

Multivariate analytic function property

Suppose function $F: \mathbb{C}^n \to \mathbb{C}$ is analytic everywhere and in every coodinate; i. e. for any $q \in \mathbb{C}^n$ and for any $j \in \{1,2,\ldots,n\}$ function $f_{q,j}: \mathbb{C} ...
3
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1answer
185 views

How to prove the identity $\pi^{s/2}=e^{(\log(2\pi)-1-\gamma/2)s}\prod_{\rho}e^{s/\rho} $?

In the wikipedia the Hadamard product for the Riemann's zeta function has two forms. The first one is ...
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6answers
378 views

The notion of complex numbers

How does one know the notion of real numbers is compatible with the axioms defined for complex numbers, ie how does one know that by defining an operator '$i$' with the property that $i^2=-1$, we will ...
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0answers
151 views

Is this a valid proof of the orthogonality of harmonic conjugates?

My textbook (Churchill) is asking me to prove that the contours $u(x,y) = c_1$ and $v(x, y) = c_2$, where $u$ and $v$ are the real and imaginary components of an analytic function $f(z)$, are ...
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1answer
78 views

Let $f(z)=e^x + ie^{2y}$ where $z=x+iy$. Where does $f'(z)$ exist?

Let $f(z)=e^x + ie^{2y}$ where z=x+iy is a complex variable defined in the whole complex plane. a)Where does f'(z) exist? b) Where is f(z) analytic? Answer: a) I used the Cauchy Riemann to test ...
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1answer
68 views

Tricky question regarding holomorphy

Question: Find all holomorphic functions $f(z)$ on $C \setminus \{0\}$ such that $$f(1) = 1,\ \ \ \ \ \ \ |f(z)| \le \frac{1}{|z|^3}$$ Attempt at solution: I've discovered that $f(z) = ...
0
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1answer
84 views

Two reflections will evidently result in a (fractional) linear transformation

I'm having hard time figuring out the following sentence in my textbook. "Two reflections will evidently result in a (fractional) linear transformation" I'm confused because I don't know if the two ...
2
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1answer
173 views

Definite integral involving hyperbolic cosine

I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this ...
3
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2answers
148 views

Computation of a certain integral

I would like to compute the following integral. This is for a complex analysis course but I managed to around some other integrals using real analysis methodologies. Hopefully one might be able to do ...
2
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1answer
171 views

The differentiability of $f(z)$ if $ \lim\limits_{z \to z_0} \frac {|f(z)-f(z_0)|}{|z-z_0|}=k$?

Let $f:\Omega \to \mathbb C$, where $\Omega$ is a region in $\mathbb C$, and $z_0 \in \Omega$. Suppose that $\displaystyle \lim_{z \to z_0} \frac {|f(z)-f(z_0)|}{|z-z_0|}=k$ for some constant k ...
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0answers
58 views

If $h(k)$ preserves the complex conjugation property, does $ikh(k)$ preserve it too?

My question is this: if $h(k)$ preserves the complex conjugation property (in other words, $h(k) = h(-k)$, $k$ can be just $-n$, $-n+1$, ..., $0$, $1$, ..., $n-1$),then $ikh(k)$ also preserves complex ...
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1answer
39 views

A simple computation of log function

I just want to make sure I got the right calculation. $$\log[(1+i)^{2i}]=\log[e^{i\ln2-\pi/2-4k\pi}]=i\ln2-\pi/2-4k\pi=i\ln2-\pi/2.$$
2
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2answers
257 views

Locate the singular points and state whether it is a pole, a removable singularity, or an essential singulatity: $z/(e^z-1)$

Locate the singular points and state whether it is a pole, a removable singularity, or an essential singulatity: $$f(z) = \frac{z}{e^z - 1}.$$ I obtained $z=0$. But I don't understand how to ...
0
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1answer
103 views

what is the difference between the Argument Principle and Rouché's Theorem

What is the difference between the Argument Principal and the Rouche's Theorem. I am not sure when to use which one when I have a question for example: How many roots of $z^4 + z^3 + 1=0$ lie ...
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0answers
66 views

Type of isolated singularity of holomorphic funtion unique?

I am using "Real and complex analysis" by rudin. I have just read the proof concerning the three different types of isolated singularities, and that one of them must occur. The book doesn't say ...
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4answers
315 views

Why $\displaystyle f(z)=\frac{az+b}{cz+d}$, $a,b,c,d \in \mathbb C$, is a linear transformation?

Now I'm confused with what "a linear transformation" means. In linear algebra textbook, I learned that a linear transformation is $T:V \to W$, where V,W are vector spaces, which satisfies additivity ...
3
votes
2answers
51 views

Is $f(z)=z^n+nwz$ one-to-one?

Let $f(z)=z^n+nwz$ be a complex function with $|w|=1$ and $n>1$ a natural number. Is this function one-to-one inside the unit circle ($|z|<1$)? ATTEMPT I didn't have a lot of luck checking ...
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1answer
133 views

Prove Complex Function Is Holomorphic

Prove for $a\gt0$ that following series is holomorphic $$ \sum_{n=1}^\infty \frac {1}{(a+n)^z} \quad \textrm{for} \quad \operatorname{Re}z \gt 1 $$ I'm trying to prove this given that $Re \quad z ...
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0answers
125 views

Need some help with my Complex Analysis textbook

I'm confused with a proof in "Complex Analysis" by Ahlfors.(P.73 2.3 comformal mapping) I need some help for the last part of the proof. "Let $f:\Omega\to \mathbb C$ and fix $z_0\in \Omega$. ...
3
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2answers
390 views

Non-existence of a bijective analytic function between annulus and punctured disk

Suppose $A=\{z\in \mathbb{C}: 0<|z|<1\}$ and $B=\{z\in \mathbb{C}: 2<|z|<3\}$. Show that there is no one -to-one analytic function from A to B. Any hints? Thanks!
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0answers
181 views

Problem calculating with the residue theorem

I've come across this integral and I'm having some problems with it. I get to a solution, but looks a bit weird and I may be doing something wrong. $\int_C\cos(e^{(1/z)})dz $ Being $C$ the unit ...
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2answers
1k views

How to find all Laurent series for $\frac{1}{z^2 - z}$ centered at $1$?

I have two questions. The definition I learned defined a laurents for a function analytic in some annalus? I guess this question is asking annali centered at $1$? How do I determine all the ...
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1answer
96 views

Can we give a definition of the cotangent based on a functional equation?

I've recently learned that the cotangent satisfies the following functional equation: $$\dfrac1{f(z)}=f(z)-2f(2z)$$ (true for $f(z)\neq 0$). Can we solve this equation for real or complex ...
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0answers
503 views

Show that the partial derivatives of a harmonic function fit into a holomorphic function

Let G be a map in $\mathbb C$ ( $G \subseteq \mathbb C $) and $u: G\mapsto \Bbb C$ is harmonic function. Then show that $f: G\mapsto \Bbb C$ $$ f(x+iy)=\frac{du}{dx} - i\frac{du}{dy}$$ is ...
6
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1answer
195 views

How to find the number of roots using Rouche theorem?

Find the number of roots $f(z)=z^{10}+10z+9$ in $D(0,1)$. I want to find $g(z)$ s.t. $|f(z)-g(z)|<|g(z)|$, but I cannot. Any hint is appreciated.
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1answer
281 views

How to calculate this integral using residue theorem?

$f,g$ are holomorphic in $D(0,1)$. $P_1,P_2,...,P_k$ are roots of $f$ in $D(0,1)$. their orders are $n_1,...,n_k$. Compute $$\frac{1}{2\pi i}\oint_\gamma\frac{f'(z)}{f(z)}\cdot g(z)dz.$$ Using ...
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1answer
200 views

Is there a conformal self-map on the upper half-plane that swaps two points?

Again, is there a conformal self-map that interchanges two points in the upper half-plane? I'm beginning to think this isn't so. Such a map would be a FLT $\frac{az+b}{cz+d}, ad-bc=1$, with real ...
2
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1answer
152 views

problems related to poles and essential singularity

I was thinking about the problem which says: Let $f(z)$ be a function defined by: $$f(z)=\frac{\sin(1/z)}{z^{2}+11z+13}.$$ Then which of the following is correct? (a) No singularity. (b) Only ...
3
votes
2answers
229 views

Proving that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \frac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$

I want to show that $\sum_{|j| < n} (n-|j|) \exp(ij\lambda)= \dfrac{\sin^2(\frac 1 2 n\lambda)}{\sin^2(\frac 1 2 \lambda)}$. I know from Proving ...
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0answers
315 views

Taylor Series of the Complex Log and Contour Integration

Write the Taylor series of $\text{Log}(1+w)$ with center at $w=0$ on $|w|<1$; check that if $|z-2|<1$, then $|z|>1$. (If you have difficulties in checking this formally, try to draw a ...
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1answer
96 views

The problem and definition of principal part

In James Brown's complex variables and applications, there is an exercise: Let $f \left( z \right) = \frac{8 a^3 z^2}{\left( z^2 + a^2 \right)^3}$ with $a > 0$. Show that the principal part of $f ...
3
votes
1answer
702 views

Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer.

Using residues, try the contour below with $R \rightarrow \infty$ and $$\lim_{R \rightarrow \infty } \int_0^R \frac{1}{1+r^n} dr \rightarrow \int_0^\infty \frac{1}{1+x^n} dx$$ I've ...
1
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1answer
281 views

Given two fixed points on unit disk,find analytic functions from unit disk to unit disk that maximize the distance between values at the two points

Given $z, w \in D$ (unit disk, open), what are the functions (analytic, from unit disk to unit disk) $f$ that maximize the norm of $f(z)-f(w)$? My attempt: We have that $$|f(z)-f(w)| \leq ...