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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Cut a hole in a Riemann surface

Consider $R$ a Riemann surface of genus $g\ge 2$ with no border component. Suppose we want to cut a hole in $R$: for example consider $J\subset R$ an embedding of the interval $[0,1]$ and then cut a ...
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1answer
9 views

Branch points+cuts, $(z+k)^{1/2}$ and $(z+k)^{-1/2}$

Do the complex functions $f(z)=(z+k)^{1/2}$ and $f(z)=(z+k)^{-1/2}$ have the same branch points? If so, why? Also, would this mean that we can take the same branch cut for both functions? Thanks.
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38 views

Why does $\sum_{-\infty}^{\infty}\frac{1}{z-n}$ diverge while $\sum_{-\infty}^{\infty}\frac{1}{z-n} + \frac{1}{n}$ converge?

Why is it that $$\sum_{-\infty}^{\infty}\frac{1}{z-n}$$ diverges while $$\sum_{-\infty}^{\infty}\frac{1}{z-n} + \frac{1}{n}$$ converges? These are both the series representations of $$ \pi ...
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1answer
25 views

Find $\int_{|z|=r}\frac{|dz|}{|z-a|^2}$ where $|a| \neq r$

Trying to find $$\int_{|z|=r}\frac{|dz|}{|z-a|^2}$$ where $|a| \neq r$. Was trying to use the maximum length estimate. For both case $|a|>r$ and $|a|<r$, I got the same answer zero.
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13 views

Limit Point and Sequences Theorem [on hold]

I am struggling with proving the following statement: Suppose f(z) is analytic in S and suppose there is a sequence of points zn element S, with a limit point in D such that f(z)=c. Then f(z)=c in ...
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2answers
12 views

Given that $( A_n Z_n )_{n \leq 0}$ converges in $\mathbb{C}$, prove that $\sum (A_n - A_{n+1}) Z_n$ converges iff $\sum A_n (Z_{n} - Z_{n-1})$ does

Starting with left implies right. I want to see that given $\epsilon > 0 $ there's an $n_0 \in \mathbb{N}$ so that for all $i$ , $j \in \mathbb{N}$ with $i > j$ $$ | \sum_{n=1} ^i a_n (Z_n - ...
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15 views

Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane

I am trying to find the Conformal Map from Domain $D$(where $D$ is bounded by $\{z=x+iy,y=1/x,x>0\}$, $2+2i\in D$) to upper Half Plane $\mathbb{H}:=\{w:Im(w)>0\}$, I am using $z+1/z$ map but ...
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2answers
17 views

Let $f(z) = \frac{\pi^2}{\sin^2(\pi z)}$. For some $z=n$, show it has a pole of order 2.

Let $$f(z) = \frac{\pi^2}{\sin^2(\pi z)}.$$ For some $z=n$, show it has a pole of order 2. Aside from appear elsewhere, this function is discussed on p188 of Ahlfors. I can answer the ...
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28 views

Fundamental Solution Cauchy-Riemann Operator

Consider the linear partial differential operator $L=\partial_{x}+i\partial_{y}$ acting on distributions on $\mathcal{D}'(\mathbb{R}^{2})$. Folland writes that a fundamental solution of this operator ...
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27 views

Find an analytic continuation

Let $f(z)=\sum_{j=0}^{\infty}z^j$ for $|z|<1$. For what values of $\alpha$ ($|\alpha|<1$) does the Taylor expansion of f(z) about $z=\alpha$ yield a direct analytic continuaton of f(z) to a disk ...
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1answer
14 views

Prove that if a function has an isolated singularity at $z_0$, then its derivative also has an isolated singularity at $z_0$

Prove that if a function has an isolated singularity at $z_0$, then its derivative also has an isolated singularity at $z_0$. Find $Res(f',z_0)$. My approach: Suppose $f$ has an isolated singularity ...
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1answer
20 views

Analyticity and differentiability of complex functions

I understand what analytic functions are and what differentiability of a complex function means but I have been reading "advanced engineering mathematics by kreyszig" and it says that the concept of ...
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0answers
16 views

A analytic function has a pole if $n ≤ |f(1/n)| ≤ n^{(3/2)}$ and $z^2 f(z)$ bounded

Let $f : \{z ∈ \mathbb{C} | 0 < |z| < 1 \} \to \mathbb{C}$ be analytic such that $n ≤ |f(1/n)| ≤ n^{(3/2)}$ for $n = 2, 3, . . ..$ Assume that $z^2f(z)$ bounded in $|z| < 1.$ Then I need ...
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1answer
18 views

Complex Function Identities

Newcomer to Complex Analysis, I can't see any reason why these identities wouldn't hold, if taking multi valued log and exp the whole time. Am I correct?
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18 views

If $f$ is Holomorphic and non zero then so is $f'/f$ on some region $R$

In my notes I am told to take this as true without proof, which is cool... for some. $f^{-1}$ is holomorphic by a property of holomorphic functions. To show that $f'/f $ is Holomorphic, I think I ...
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3answers
28 views

showing that $\int_\gamma f=0$

Let $G \subset \mathbb{C}$ be an open star domain with the vantage point $z_0$, let $f: G \rightarrow\mathbb{C}$ be holomorphic and let $ \gamma: [0,1] \rightarrow G$ be a closed $C^1$-curve. I got to ...
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0answers
18 views

On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
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14 views

Show that this is an entire function [duplicate]

If $f$ is an entire function, I have to show that $g(z):= \overline{f(\overline {z})}$ is also an entire function. How can I proof this if $g:\mathbb{C}\to\mathbb{C}$?
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2answers
23 views

Image of the set $\{z \in \mathbb{C}:|z|>r\}$ by the complex exponential function

I need to find the image of the set $\{z \in \mathbb{C}:|z|>r\}$ by the complex exponential function $x+iy \rightarrow e^x(\cos(y) + i \sin(y))$. Previously, I found that the image of horizontal ...
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1answer
21 views

Find the real and imaginary parts of $f(z)=\frac{2}{z+1}$

Find the real and imaginary parts of $f(z)=\frac{2}{z+i}$. Denote the conjugate of $z$ as $\bar{z}$, then ...
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1answer
66 views

The doubly infinite series $\sum_{-\infty}^{+\infty} n$

I have the following question from Function Theory of One Complex Variable - Greene/Krantz: Give an example of a series of complex coefficients $ a_n$ such that $\lim_{N \to + \infty} \sum_{n= ...
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1answer
15 views

Residues of the product of the gamma and zeta functions.

If we have a function of the form $f(z)=\Gamma(z)\zeta(z)$ where $\Gamma(z)$ is the standard gamma function, and $\zeta(z)$ is the Riemann zeta function, then how would we calculate the residues? I ...
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0answers
10 views

How to prove the Mellin's inverse formula?

Let $f:[0,+\infty)\to \mathbf{C}$ be a continuous, compactly supported function. Establish the Mellin's inverse formula $$f(u)=\frac{1}{2\pi i}\lim_{T\to \infty}\int_{c-iT}^{c+iT} ...
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1answer
26 views

How to classify singularities of complex functions

I have been trying to understand complex analysis but can't figure out how to classify singularities of a function. For example I am stuck on the following question: Classify the singularities of the ...
3
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1answer
49 views

Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$

I've been given a proof that shows the following: If $f:[a,b]\to \mathbb C$ is a continuous function and $f(t)=u(t)+iv(t)$ then $$\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| ...
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1answer
39 views

Evaluate $\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$ using complex integration

I'm trying to evaluate the real integral $$\int_{ - \infty}^{\infty} \frac{dx}{1+x^2}$$ Denote $\Gamma_{1}=\left[-R,R\right]\ \Gamma_{2}=Re^{it}$, for $t\in\left[0,\pi\right]$, and let $\gamma$ be a ...
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1answer
38 views

Math-english for non-natives: What does “supported in” mean?

As a non-native English speaker, I am struggling with the following sentence: "Fix a function $f:\mathbb{R}\to\mathbb{C}$ such that $f$ is supported in the unit Ball." Does this mean ...
3
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1answer
53 views

Show that $\frac{z^2}{z-3}$ is analytic.

Explain why $\displaystyle \int_{C_1(0)} f(z) dz =0$ for the function $\dfrac{z^2}{z-3}$. In case there's some confusion with the notation, $C_1(0)=$ circle of radius $1$ centered at $0$ in ...
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0answers
20 views

Describe the image of $|z-2i|=2$

Describe (as fully as possible) the image of $|z-2i|=2$ under the function $f(z)=\frac 1 z$. I'm not actually sure what this is asking, but I did the mapping into the w plane and came up with $2iw ...
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1answer
31 views

solve $\sin(z)=-1$ in the set of complex numbers

I'm pretty sure I'm on the right track, but am I missing anything? Can anything further be done with this? Solve $\sin(z)=-1$ in the set of complex numbers. $\sin(z)=-1$ $\Rightarrow{e^{iz}-e^{-iz} ...
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0answers
18 views

Can Hecke Operators be defined on more general spaces of elliptic curves?

Classically, the Hecke Operators act as endomorphisms of $\omega^k_{\mathcal{M}_{ell}(\mathbb{C})}$, defined by noting that there is a distinguished class of covering maps $\widetilde{E}\to E$ given ...
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1answer
22 views

Why can you replace $\sin(az)$ with $\exp(iaz)$ in $\int_{-\infty}^\infty \frac{\sin(ax)}{x(x^2+1)}dx$?

I am looking at the exercise VII.5.2 from Gamelin's Complex Analysis: Show using residue theory that $$\int_{-\infty}^\infty \frac{\sin(ax)}{x(x^2+1)}dx = \pi(1-e^{-a}),\quad a>0.$$ ...
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0answers
14 views

Show there is no intersection between sequence of balls on an open set.

Is there a way to show that a finite sequence of balls, in an open set $U ⊆ C$, with a small enough radius's such that there is no intersection between any of the balls? Intuitively it makes sense to ...
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0answers
8 views

Find the most general bilinear transformation that maps $|z-1|=1$ to $\Re(f(z)) = 1$.

Find the most general bilinear transformation that maps $|z-1|=1$ to $\Re(f(z)) = 1$. Attempt Assuming that $$ 2 \mapsto 1 \\ 0 \mapsto \infty \\ 1+i \mapsto 1+i $$ we have the bilinear ...
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0answers
14 views

Taylor series roots at infinity

I started thinking about this after this MathSE thread. Take a sequence of Taylor polynomials $f_n$ that converge to $f$. Does $f_n$ always have a growing number or roots in $\mathbb{C}$ which grow ...
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16 views

On spectrum of periodic boundary value problem

Consider the following boundary value problem on the infinite strip $(-\infty,\infty)\times[0,1]$ w/periodic conductivity $\gamma(x,y)=\gamma(x+2\pi,y)>0$: $$\begin{cases} ...
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31 views

Roots of the Taylor approximation of the exponential

While answering another question, I looked at the roots of the $n^{th}$ degree Taylor approximation of the exponential. $$e^x\approx E_n(x):=\sum_{k=0}^n\frac{x^k}{k!}.$$ Apparently, these root are ...
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Complex analysis techniques in fluid dynamics [on hold]

What are some of the areas in which complex analysis methods are applied in fluid dynamics?
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39 views

If $f(z)$ is analytic how about $\overline{f(z)}$ and $f(\bar{z})$?

Are they generally not or are they always not? How can it be proven that they're not?
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11 views

Applying the Schur algorithm to finite Blaschke products

A Schur function is a function which is holomorphic in the unit disk $\mathbb{D}$ satisfying $|f(z)|\leq 1$. The Schur algorithm is a way of producing a sequence of Schur functions starting with a ...
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2answers
19 views

computation of a line integral

in an excersise I have to compute a few line integrals but one of them I can't solve. It is not even written as a line integral but the others are. I am talking about: ...
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1answer
18 views

showing that the Cauchy-Riemann equations are satisfies at a point not in the domain of some function $f(z)$

Let's say I have a function not defined at $z=0$ and thus written as a composite function. That is, something in therms of $x$ and $y$ for $z \neq 0$ and $0$ for $z=0$. To verify that the ...
3
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1answer
25 views

Use Rouche's Theorem to find the radius of the roots of the following polynomial

Using Rouche's Theorem, find the radius of the roots of $z^{3}+z^{2}-2z+3$ To answer this question, I have set functions $f\left(z\right)=z^{3}$ and $g\left(z\right)=z^{2}-2z+3$, to try and show ...
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2answers
20 views

Simplyfing complex expressions with square roots

Simplify the expression $z =(4+4 \sqrt 3 i)^{1/2}$ so that it's in the form $z = x + iy$. So far I got: $$4(1+ \sqrt3 i)^{1/2}$$ But I'm unsure where to go next. I don't know you how can ...
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0answers
9 views

Definition: When two lattices are similar?

I have a question where I need to show two lattices are similar. But I don't know the definition and I can't find on Internet. Thanks
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3answers
36 views

Complex analysis: Using Taylor expansion to show $|c_n| ≤ \frac{1}{r^n}\sup_{z∈C_r(0)}|f(z)|$

Consider the function $f$ is defined through the power series $$f(z) := c_0 + \sum_{n=1}^\infty c_nz^n$$ and assume that the series on the right has a radius of convergence $R > 0$. Show that ...
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1answer
16 views

Showing complex transformations in a fluid way

It says it all in the title: I need to show how simple complex transformations (translations and dilations, or even both) affect shapes on the complex plane in a "fluid" way – that is, creating some ...
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17 views

Alternative way of finding a logarithm branch of a polynomial in a certain set

First the question: Let $$r(z) = a_n + a_{n-1}z^{-1} + ... + a_0z^{-n}$$ I need to show that there exists a $K>0$ so that there exists a branch of $log(r(z))$ in $D = \{ |z| > K \}$. I know ...
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0answers
18 views

Proof of a sequence of holomorphic functions

How can I prove with using estimates of Cauchy: a sequence of holomorphic functions {$f_n$} uniformly converges to the holomorphic function f on D $\subset$ $\mathbb{C}$. Then for each compact set K ...
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0answers
28 views

show the complete Proof of the proposition 4.3.6 from the book “Basic Complex Analysis” [on hold]

Proposisition 4.3.6: (i) Suppose $f$ is analytic on an open set containing the closed upper half plane D ={ $z \in C | Im(z)\ge 0$} except for a finite number of isolated singularities none of ...