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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1answer
422 views

Conformal map from exterior of unit circle to upper half plane

I'm trying to find a conformal map from the space $\Omega = \mathbb{H}\setminus\{z : |z-\frac{i}{2}|\leq\frac{1}{2}\}$ to the upper half plane. I think I'm most of the way there, but I wanted to check ...
2
votes
0answers
182 views

Coefficients of the Weierstrass $\wp$'s Laurent expansion

I am trying to study the Weierstrass $\wp$ function using a combination of texts from Alfors; Cartan; Freitag and Busam; and Siegel. But I am having some trouble because I would like to try to avoid ...
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0answers
32 views

Estimation Question

Suppose $f: A \to \mathbb{C}$ is analytic on an open set $A$ containing the closed half plane $H=\{z \in \mathbb{C}: Im(z) \ge0 \}$ and that there is a finite constant $M$ with $|f(z)| \le M$ for all ...
0
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1answer
251 views

About the branch-cut in the complex logarithm

Say I have the function $log (f(z))$, then does the imaginary part of the value go down by $-i 2 \pi$ on crossing the branch-cut of the log function even if $f(z)$ is not crossing the branch-cut? ...
1
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1answer
59 views

Showing reflections are hyperbolic isometries in $\mathbb{D}$.

I am interested in showing that isometries in $\mathbb{D}$ are either conformal self-maps in $\mathbb{D}$ or they are compositions of conformal self-maps with $z\mapsto \bar{z}$. It is given that ...
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votes
1answer
105 views

open mapping theorem problem

Here is the question I need your help. Let $f$ be an analytic on $U=D(z_0,R)\setminus{z_0}$ such that $z_0$ is a pole of $f$. Prove that for any $r\in (0,R)$ there is $m \in (0,\infty)$ such that ...
4
votes
5answers
180 views

Evaluating $\int_0^{2 \pi} \sin^4 \theta\: \mathrm{d} \theta$

Evaluate the following integral: $$\int_0^{2 \pi} \sin^4 \theta \:\mathrm{d} \theta$$ My approach: Parametrize and obtain $$\frac{1}{(2i)^4} \int_{|z|=1} \left (z-\frac{1}{z} \right)^4 ...
2
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1answer
73 views

Three questions concerning holomorphic functions defined by contour integrals

Consider the following situation: a simple, closed, piecewise smooth curve $\gamma$ in the complex plane and $\Omega$ the bounded connected component of the complement of $\gamma$ in $\mathbb{C}$; a ...
1
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1answer
117 views

Schwarz-Christoffel transformation understanding

I've been reading this explanation (with pic and formula) about the Schwarz-Christoffel mapping. I'm not really used to this sort of argument. My question is why are all terms constant in $(21.3)$ ...
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0answers
55 views

Conditional convergence of $\sum _{n=1}^{\infty} a_n$ and $\sum _{n=1}^{\infty} \log(1+a_n)$

In the Ahlfors Complex Analysis book section on infinite products, there is a result that the the series $\sum _{n=1}^{\infty} |a_n|$ converges exactly when $\sum _{n=1}^{\infty} | \log(1+a_n)|$ does. ...
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1answer
348 views

f is an entire function, show that it is a constant function using Liouville theorem

let f be an entire function such that $|f(z) + e^z| > |e^zf(z)|$ for all z in C. Show that f is a constant function. Suggestion is to use Liouville's theorem then show that f is actually the ...
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0answers
48 views

Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
0
votes
1answer
71 views

Does this conformal map from a rectangle exist?

It is well known (by Schwarz-Christoffel) that if $k \in (0,1)$, then the Jacobi elliptic function $\mathrm{sn}(\cdot,k)$ provides a biholomorphic map from the rectangle $(-K(k),K(k)) \times ...
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1answer
58 views

Show uniform convergence of a series of complex function on every compact subset

Let $f:B(0,1)\rightarrow \mathbb{C}$ be an analytic function. Suppose $\sum^\infty_{n=0}f^{(n)}(0)$ converges absolutely. Show that there exists an entire function $g(z)$ such that $g(z)=f(z)$ for ...
1
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1answer
36 views

Do these imply the function is Analytic?

Let $f(x,y) : U \rightarrow \mathbb C$ such that $ \partial f/ \partial y$ and $ \partial f/ \partial x$ are continuous and $\forall c \in U$. Let $ S_c$ = circle centered at $c$ lying inside $U$ and ...
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1answer
42 views

What's an importance of multi-valued functions?

For example, let's define $\log z= \{w\in \mathbb{C} : e^w=z\}$. Then, we call this set $\log z$ a "multi-valued function". Formally saying, this $\log z$ is merely a set but not a function to ...
2
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1answer
73 views

The fixed points of analytic self-maps of $\mathbb{D}$

So far, I have assumed that $z_1$ is a fixed point of an analytic self map of $\mathbb{D}$. Then, I summoned the conformal self map of $\mathbb{D}$, $\phi$ to take $z_1\to 0$. It follows from Schwarz ...
13
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1answer
254 views

Proving $\left(\sum_{n=-\infty}^\infty q^{n^2} \right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi \tau)}$

The so-called "two squares theorem" can be proven by establishing the following identity: $$\left(\sum_{n=-\infty}^\infty e^{\pi i \tau n^2}\right)^2 = \sum_{n=-\infty}^\infty \frac{1}{\cos(n \pi ...
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0answers
27 views

About analytic functions with nonnegative real part in the right-hand plane

I am reading some lecture notes where the following claim is made. Suppose $h(s)$ is analytic in the region $\{{\rm Re}(z) > 0 \}$, $h(\omega) \in \mathbb{R}$ for all $\omega \in \mathbb{R}$, and ...
4
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1answer
92 views

Show $f$ is a complex polynomial of degree at most 2

Suppose $f:\mathbb{C}\rightarrow\mathbb{C}$ is an entire function and $$\displaystyle\min\{\left|f'(z)\right|,\left|f(z)\right|\}\leq \left|z\right|+2$$ for all $z\in\mathbb{C}$. How to see that ...
0
votes
2answers
68 views

Understanding the Definition of $\int_\gamma f\ \overline{dz}$

Definitionally, we have that $$ \int_\gamma f\ \overline{dz} = \overline{\int_\gamma \overline{f}\ dz} $$ Now let $\int_\gamma f\ dz = w = x +yi$. Question 1: Is it not the case that $\int_\gamma ...
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1answer
19 views

Defining the Complex Line Integral w.r.t. $x$ and $y$

Ahlfors defines line integrals with respect $x$ as follows: $$ \int_\gamma f\ dx = {1 \over 2} \left( \int_\gamma f\ dz + \int_\gamma f\ \overline{dz} \right) $$ From this I take it as obvious that ...
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1answer
44 views

Show $f(z)=i$ given following conditions

Suppose $f(z)$ is a polynomial of degree at most 1. $f(0)=i$ and $|f(z)|^2\leq 1+|z|^3 \;\forall z\in \mathbb{C}$. Show $f(z)=i$ for all $z\in \mathbb{C}$. We may let $f(z)=az+i$ for some $a\in ...
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1answer
104 views

finding pole of order m of a function

Please help me with this question: Suppose that a function $f$ has a pole of order $m$ at $z = 0$, and $\displaystyle\frac{1}{\displaystyle|z|^\frac{3}{2}} \le |f(z)| \le ...
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1answer
25 views

Complex Modulus Is Less Than One

Show that $|\frac{-a}{b}+\frac{\sqrt{a^2-b^2}}{b}|<1$ where $a>|b|>0$ This is just a minor part of a more full complex trigonometric integral proof that I'm working on. I'm pretty much set ...
0
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1answer
25 views

Complex Line Integral(Meromorphic)

I used partial fractions to separate the denominator but I can't figure out what the purpose of splitting the function into two integrals is. I don't think they are analytic continuations of the other ...
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0answers
48 views

What are some good general estimates?

For example, the triangle inequality for complex numbers and summations is a good one. Also, the ML-Estimate (Estimation Lemma), Cauchy Estimates $|zw|=|z||w|$. As you can probably notice, I really ...
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2answers
291 views

How to find the complex roots of unity in polar form quickly

Imagine we want to find fourth complex roots of unity of $z=-16$. We first write the number in polar form: $z=16e^{i\cdot\pi}$ Then we use DeMovier theorem to find a fourth root as follow: ...
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0answers
40 views

Verify that the identity $Log(\dfrac{1+z}{1-z}) = Log(1+z) - Log(1-z)$ holds when $|z|<1$.

Verify that the identity $$Log(\dfrac{1+z}{1-z}) = Log(1+z) - Log(1-z)$$ holds when $|z|<1$. The procedure on the book is as follows: \begin{align*} Log(\dfrac{1+z}{1-z}) &= ...
0
votes
1answer
32 views

Linear indepedent holomorphic functions

Suppose you have a given set of holomorphic functions $e_\alpha(z)=\exp(\alpha_1z_1+\dots+\alpha_nz_n)$ for different $\alpha=(\alpha_1,\dots,\alpha_n)$ in an open set of $\mathbb{C}^n$. How can I ...
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0answers
37 views

Does anyone know an example of a local homeomorphism from the open unit disc onto itself that is not a homeomorphism? [duplicate]

I am interested in an example of a local homeomorphism from the open unit disc D onto itself which is not a homeomorphism. Or, could one prove that any such map is a homeomorphism?
4
votes
0answers
99 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
0
votes
2answers
50 views

Let $f(z) = \sum_{j=0}^{\infty}a_jz^j$ be the Maclaurin expansion of a fnction $f(z)$ analytic at the origin. Prove each of the following statements.

Let $f(z) = \sum_{j=0}^{\infty}a_jz^j$ be the Maclaurin expansion of a function $f(z)$ analytic at the origin. Prove each of the following statements. $(a)$ $\sum_{j=0}^{\infty}a_jz^{2j}$ is the ...
1
vote
1answer
39 views

Finding branches at which an identity holds

If $\log_\theta$ is the branch $\log_\theta=ln(r)+i\Theta(r>0, \theta <\Theta<\theta +2\pi)$ for which $\theta \in [0,2\pi]$ does $\log_\theta(i^2)=2\log_\theta(i)$ hold. Evaluating these ...
0
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1answer
47 views

Show two complex functions are conjugate

I am stuck on a homework problem that asks Show that the functions $f(z) = \frac{z^2}{z^2 + 1}$ and $g(z) = z^2 + 1$ are conjugate. Two functions $f$ and $g$ are conjugate if there is a ...
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1answer
47 views

Integrating a complex function with Cauchy formula

We have I =$\oint_{C}^{} \frac{(z-1)\sin(z)}{z^2 - 2z - 3}$, C is a circle for which $|z-2| = 2$. I wrote $I = \oint_{C}^{} \frac{(z-1)\sin(z)}{4(z-3)} - \oint_{C}^{} \frac{(z-1)\sin(z)}{4(z+1)}$ ...
1
vote
1answer
234 views

Complex Analysis - Definition of Singular Point

I have been reviewing Dennis Zill's Complex Analysis text and he defines a singular point as a point $z$ at which a function $f$ fails to be analytic. Now he goes on to talk about isolated ...
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2answers
111 views

Integration $\int_{-i\infty}^{i\infty}{\frac{a^{z+1}}{z+1}}\operatorname d\!z$

So, I'm trying to evaluate the integral below, and I'm having a very difficult time even getting started. $$\int_{-i\infty}^{i\infty}{\frac{a^{z+1}}{z+1}}dz$$ where a is a real number, and I am to ...
2
votes
2answers
62 views

Radius of convergence two power series (by using Cauchy test).

Let power series $\sum_{n=0}^{\infty} a_nz^n$ have radius of convergence $R$. I would ask you, is it true that $\sqrt[n]{a_n} \rightarrow \frac{1}{R}$? If it is true, then power series ...
2
votes
0answers
66 views

Why does the space of germs construction correspond to the gluing construction of Riemann surfaces?

I know this might be too broad / vague a question, but still looking for somebody to write something meaningful about this. When constructing the Riemann surfaces, why does the space of germs ...
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1answer
50 views

Power series of $f(z) = \frac{z}{1-z}$

Find power series of $f(z) = \frac{z}{1-z}$ in point $z_0 = i$ and find radius of convergence this power series. Of course, I can find $f^{(n)}(z_0)$ and then I will have $$f(z) = \sum_{n=0}^{\infty} ...
0
votes
2answers
92 views

A simple yet complex path integral

Let L be an elipse arc with parametrization $z = 2 \cos(t) + 4i \sin(t)$, $t \in [0, \frac{\pi}{2}]$. How would one solve $\int_{L}^{} z^{-1} dz$?
2
votes
1answer
151 views

Application of Riesz representation theorem

Suppose the following situation. We have linear functional $l$ on the space $H(\mathbb{C}^n)$ of entire function and wish to find a representation for $l$ with integration against a complex Borel ...
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1answer
112 views

$v$ is Conjugate Harmonic to $u$ $\implies$ $f = u + iv$ is Analytic (Proof Verification from Ahlfors)

Hypothesis: Let $u$ and $v$ be two functions from $\mathbb{R}^2$ to $\mathbb{R}$ s.t. $$ \Delta u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0 $$ and $$ \Delta v = ...
0
votes
2answers
73 views

Complex Number Question - $|z^{z}|$

Find all possible values of $$\mid z^{z} \mid$$ using the polar for of $z$. I have tried putting it into polar form but nothing comes out that seems easy to work with/looks like a reasonable simple ...
1
vote
1answer
24 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
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0answers
55 views

Fundamental Theorem of Analysis: Help in understanding steps of proof.

Statement (Fundamental Theorem of Analysis): Let $f: I \rightarrow \mathbb C$ be continuous on $I \subseteq \mathbb R$ and let $a \in I$. Define $$F(x) := \int^x_a f(t) \, dt, \ x\in I.$$ Then $F$ is ...
1
vote
1answer
112 views

Mentally visualizing functions of complex numbers

I've recently been learning about functions of complex numbers (to complex numbers), and I can't quite fit them into my head. When I think about real functions, I tend to mentally visualize them as ...
6
votes
1answer
115 views

enitre function that preserve the rationals?

Here's a question i would be curious to know the answer The question is: what is the set of all entire functions $f: \mathbb{C} \to \mathbb{C}$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$.
0
votes
1answer
46 views

Residue of Function

I need help finding the residue of the following function at $z=\pi$: $$\exp\left(\frac{2}{z-\pi}\right)$$ I have put the function into a series expansion about z=pi by using the expansion of $e^x$ ...