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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64 views

Can I find an analytic self map of the unit disc satisfying lower conditions?

I am trying to find an analytic self map $\phi$ of the unit disc $\mathbb{D}$, which satisfies $$\phi\neq\mathrm{id};\; \mbox{and}$$ ...
27
votes
1answer
654 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
0
votes
1answer
163 views

Conformity of $r^λ(\cos(λθ),\sin(λθ))$ for $λ∈\Bbb C$, $(r,θ)$ standard polar coordinates

Welcome everybody :) I have problems to solve the following task: Let $\lambda∈\Bbb C$. Define $$(r,\theta ) = r^λ (\cos(λ\theta ),\sin(λ\theta ))$$ Here $(r,\theta )$ are the standard polar ...
0
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1answer
61 views

complex vector fields - hard d vs. soft d?

I believe this is a computation I have done before, but now I can't write the symbols to convince myself: What is the connection between the "hard" complex differential operator d/dz and the "soft" ...
3
votes
1answer
82 views

Problem with the proof that $\zeta(s)$ has no zeros for $\mathrm{Re}(s) = 1$

Almost every proof I read says that If $\zeta(s)$ has a zero of order $\mu$ in $1 + ai$ ($\mu \geq 0$) then $$\lim_{\epsilon \to 0}\; \epsilon \frac{\zeta'(1+\epsilon +ai)}{\zeta(1+\epsilon ...
1
vote
1answer
52 views

Is just continuity enough to prove this?

Sorry if that´s an idiot question. Let $f: D \longrightarrow \Omega$, such that $D$ is the unitary open disc centered at the origin and $\Omega = \{z \in \mathbb{C}; \mathscr{Re}(z) \geq 0 \}$. If ...
2
votes
1answer
45 views

Laurent series of the function $M(s) = \dfrac{(1-s)[\Gamma(s)\Gamma(1-s)]^2}{(2+s)(1+2s)(3+2s)} $

The function has double poles at $ s = 2,3,\ ... $ (and at other points as well but I am interested only at these points.) Its given that the principal part of Laurent series at these points $ s = n, ...
2
votes
1answer
92 views

equality of complex integrals

Good day! Help me understan what's I'm wrong Consider a function $f$ that is holomorphic in the unit disk $|z|\le 1$. Prove that $$ \int\limits_{0}^{1} f(x)\,dx = \int\limits_{|z|=1} f(z) \log ...
0
votes
1answer
92 views

a question on Cauchy's integral

I know from calculus that, if $C$ is closed path in the complex plane and $a$ a point inside $C$, then $\int_C \frac{dz}{z-a}=2\pi i$ What happens if $a$ is not inside. Is it true that $\int_C ...
1
vote
1answer
50 views

Uniform Rectifiability

What is the definition of uniform rectifiability as used in the context of analytic capacity of compact sets in $\mathbb{C}$? The precise context is this paper by Mattila, Mernikov and Verdera.
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0answers
53 views

What exactly does it mean to say that “functions cannot be integrated on Riemann surfaces”?

I've seen statements of this sort used to motivate the introduction of differential forms, and I'm not sure exactly what's meant. Obviously if you start by defining differentiation as an operation ...
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vote
0answers
279 views

Every non-constant holomorphic map of Riemann surfaces is a ramified covering

I'm reading "Riemann Surfaces" by Farkas and Kra. Section I.1.6 contains the following proposition: Let $f : M \to N$ be a non-constant holomorphic mapping between compact Riemann surfaces. ...
4
votes
1answer
366 views

How do I integrate $\int_{0}^{1}\!\sin x^2\,dx$?

How do I integrate $$ \int_{0}^{1}\!\sin x^2\,dx? $$ Will it be so complicated?
1
vote
1answer
266 views

Rudin's Construction of Lebesgue Measure

Self-studying Rudin's RCA, and I want to make sure I am understanding the intricacies of his construction of the Lebesgue measure on $\mathbb{R}^n$. The uniform continuity of $f$ shows that there ...
5
votes
2answers
109 views

Proving $g(\omega)=\frac{1}{2\pi i}\int_{\gamma}\frac{zf'(z)}{f(z)-\omega}\, dz$ where $g$ is the inverse of $f$

I have the following exercise: Let $G$ be an open subset of $\mathbb{C}$ and let $f$ be a one to one function in $H(G)$ such that $f'(z)\neq0$ for all $z\in G$. For each $\omega\in f(G)$ ...
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vote
1answer
165 views

Trying to find more information about “Darboux's method/theorem” on coefficients of an analytic function

My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down ...
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4answers
103 views

Does this limit make any sense?

Does it make sense to take the following limit? $$\lim_{\phi\to\infty}e^{i \phi}=?$$ And if yes, what does it yield? EDIT: I vaguely remember someone mentioning that this limit gives zero in a ...
1
vote
1answer
96 views

Prove there is no such analytic function

Please help prove that there is no analytic function on $z=0$ such that $$ n^{-\frac3 2}<\left|f\left(\frac1 n\right)\right|<2n^{-\frac3 2}$$ for every natural $n$.
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vote
0answers
122 views

Can we compute Fourier series of any function this way?

There is a technique to compute Fourier series much quickly, but I doubt how general this technique can be. Let's look at a simple example to see how the technique goes. Compute Fourier series of the ...
4
votes
1answer
181 views

Learning Complex Analysis: Integrals vs. Power Series - ordering the development of results.

Over the last few months, I have been visiting elementary complex analysis. My exposure to complex analysis is pretty much limited to the material in three books: Ahlfors, Bak/Newman, and ...
3
votes
2answers
304 views

Continuous extension of a Bounded Holomorphic Function on $\mathbb{C}\setminus K$

Let $f:\mathbb{C}\setminus K\rightarrow\mathbb{D}$ be a holomorphic map, where $K$ is a compact set with empty interior. My question: Prove or disprove that: $f$ extends continuously on ...
2
votes
1answer
85 views

local rotations of complex functions at roots

Suppose I have $f:\mathbb{C}\rightarrow\mathbb{C}$ with a root at $r$, and I want to make a $g(z)$ for which $r$ is also a root but $\lim_{z\rightarrow r} g(z)/f(z) =e^{i\theta}$ for some ...
2
votes
1answer
48 views

Problem with the Analytic theorem

The analytic theorem claims that If $f(t)$ is a bounded and locally integrable function on $t \geq 0$ and $g(z) = \int_0^\infty f(t)e^{-zt}$ is analytic with $\Re(z) > 0$ and extends ...
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votes
2answers
66 views

Power series in complex analysis

We derived from cauchy's integral formula that a holomorphic function converges locally in a power series. Now we had the Identity theorem and I wanted to know whether I can conclude from this that ...
4
votes
3answers
121 views

Complex Analysis Advice

Could anyone advise on this problem? Let $g(z$) be an analytic function in punctured ball $B(z_1, R) - \{z_1\}$ and let $N$ be a fixed non-negative integer such that $\lim_{z\rightarrow\ z_1}(z- z_1) ...
6
votes
1answer
298 views

Prove that $z_{n+1} = \frac12 \left(z_n +\frac{1}{z_n} \right) $ converges to $1$

For a complex sequence $\{z_n\} $ defined by the recurrence relation $$z_{n+1}=\frac12 \Bigg(z_n +\frac{1}{z_n} \Bigg) $$ where $\arg z_n \in (-\frac{\pi}{2},\frac{\pi}{2})$, prove that it converges ...
4
votes
1answer
102 views

Analytic Capacity

For a compact set $K\subset\mathbb{C}$ the analytic capacity is defined as $$\gamma(K)=\sup\{|f^\prime(\infty)|:f\in M_K\}$$ where $M_K$ is the set of bounded holomorphic functions on ...
2
votes
3answers
1k views

Express $\cos 6\theta $ in terms of $\cos \theta$

I think I'm supposed to use the chebyshev polynomials, as in $$ \cos n \theta = T_n(x) = \cos(n \arccos x)$$ But no idea what now?
1
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2answers
827 views

Why does $(\cos \theta + i \sin \theta)^n =(\cos n\theta + i \sin n \theta)$

Is it the Euler identity $$ e^{i \theta} =(\cos \theta + i \sin \theta)$$ $$ e^{i n \theta} =(\cos n \theta + i \sin n \theta)$$
2
votes
0answers
78 views

Can every complex space be covered by a finite number of Stein spaces?

Can every complex space be covered by a finite number of Stein spaces?
2
votes
1answer
70 views

Complex integral on curve

I have to show that this integral is zero, but don't know how to evaluate it. Consider a closed class $C^1$ curve $c:[a,b]\rightarrow\mathbb{C}\backslash \{0\}$ and show that $$\int_a^b\frac{\langle ...
2
votes
2answers
446 views

Prove Euler's Formula using MacLaurin Series

How can you prove $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ (Euler's Formula) using MacLaurin Series? Thanks!
0
votes
1answer
609 views

Singularities of $\cot(z) - \frac 1 z$

Define $f(z) := \frac{\cos z }{\sin z } -\frac 1z$. This is an exam-question. I have to determine singularities in $\mathbb C \cup \{ \infty \}$ and what sort of singularities they are. I further have ...
1
vote
0answers
282 views

Need help to understand branch cuts

I have a question about branch cuts. Suppose you have $f(z) = \sqrt{z^2 -1} $. Then the branch points are $ \pm 1$, so we can make a branch cut from $ (- \infty , 1]$ in order to define $f$ ...
2
votes
1answer
101 views

compute an integral with residue

I have to find the value of $$\int_{-\infty}^{\infty}e^{-x^2}\cos({\lambda x})\,dx$$ using residue theorem. What is a suitable contour? Any help would be appreciate! Thanks...
0
votes
1answer
772 views

About the existence of harmonic conjugate

I am reading Donald Sarason's "Notes on Complex Function Theory". I have two questions about the following (taken from page $88$ of the book): Why did we had to use $g$ ? We already had $f$ which ...
4
votes
2answers
130 views

First derivative of holomorphic function

I want to prove that $ |f'(z)| \le \frac{1}{1-|z|}$ where $f:B(0,1) \rightarrow B(0,1)$ is a holomorphic function. My idea was to use Cauchy's integral formula. The fact that $||f||\le 1$ might be ...
1
vote
1answer
104 views

Winding number of image curve

How many turns does $f(z) = z^{40} + 4$ make about the origin in the complex plane after one circuit of $|z| = 2$?
2
votes
1answer
162 views

Uniquely defined function

I am struggling with the following exercise: Let $f:B(0,1) \rightarrow \mathbb{C}$ be a holomorphic function and we have $\forall n \in \mathbb{N}_{\ge 2}: f'(\frac{1}{n})=f(\frac{1}{n})$ then f can ...
2
votes
2answers
79 views

For analytic $f$ on $D_2(0)$ with $|f(z)| \le |\sin z|$ on $\partial D_2(0)$ , show $|f(\frac{\pi}{2})| \le \frac{4}{\pi}$

Let $f$ be analytic on $D_2(0)$ and continuous up to the boundary with $|f(z)| \le |\sin z|$ on $\partial D_2(0)$. Prove that $|f(\frac{\pi}{2})| \le \frac{4}{\pi}$. This problem appears on an old ...
3
votes
2answers
191 views

If $u,v$ are harmonic and satisfy C-R on a set with a limit point, must $u +iv$ be analytic?

Let $\Omega \subseteq \mathbb{C}$ be a domain, and suppose that $u,v$ are real-valued harmonic functions defined on $\Omega$. Furthermore, let $W \subseteq \Omega$ be the set on which $u,v$ satisfy ...
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0answers
92 views

Finding a complex function $f$ and the residue of $f '(z)$ at $z=0$

Let $U,a$ real positive constants, $\varphi_1, \varphi_2$ $C^1$ functions on $[0,a]$ with $\varphi_1(0) = \varphi_2(0)$ and $\varphi_1(a) = \varphi_2(a)$. The problem is to find an analytic function ...
5
votes
2answers
104 views

$w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$

I am stuck on the following problem: Let $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$.Then the triangle in the complex plane with $w_1,w_2,-1$ as vertices ...
1
vote
1answer
50 views

If $|f(z)|\le1$ if $|z|\le 1$ then $|a_k|\le1$

Given the polynomial $f(z)=a_nz^n+a_{n-1}z^{n-1}+ \dots+a_0$ is bounded by $1$ on a unit disc, which means $|f(z)|\le1$ if $|z|\le 1$. Prove that $|a_k|\le1$ for all $k$. I haven't found any idea ...
1
vote
1answer
132 views

Constant function and Argument Principle

Let $h$ be a function holomorphic in the region $D\subset \mathbb{C}$. Let $C\subset D$ a Jordan curve (rectifiable) with interior $D_C$ in $D$. Suppose that for each $z\in C$, $ \operatorname{Re} ...
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vote
0answers
74 views

Let $a(z),b(z)$ be two non-zero complex polynomials such that…

I am stuck on the following problem that says: Let $a(z),b(z)$ be two non-zero complex polynomials. Then $a(z)\overline{b(z)}$ is analytic iff $a(z)$ is constant $a(z)b(z)$ is ...
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vote
2answers
62 views

The minimum possible value of $\,\,|w|^2+|w-3|^2+|w-6i|^2$

I am stuck on the following problem : What is the minimum possible value of $\,\,|w|^2+|w-3|^2+|w-6i|^2\,\,,w \in \Bbb C,i=\sqrt{-1}\,\,$? The options are $\,\,15,45,20,30.$ I have no ...
1
vote
1answer
97 views

General Möbius transformation mapping $|z|=r$ onto itself

How to find the general form of a Möbius transform that maps the circle $S=\{z\in\mathbb{C}:|z|=r\}$ onto itself.
1
vote
1answer
144 views

Limits of complex error and gamma functions in the complex plane?

What are the following one-sided limits in the complex plane (in the form $x+iy$): For the complex error function: $\lim_{x \to 0^+, y \to 0^+}\text{erf}\left(x+iy\right) = $ $\lim_{x \to +\infty, ...
7
votes
1answer
976 views

Dogbone contour integral/branch cuts/residue at infinity

I am trying to compute: $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2} dx$$ by contour integration. I define $f(z) = \sqrt{z-z^2}$ with a branch cut on $[0,1]$ in such a way that $f(-1)=\sqrt{2}i$, then define ...