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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28 views

How does one prove that two punctured disks are conformally equivalent? [on hold]

Let D1 = {z: 0 < |z| < R1} and let D2 = {z: 0 <|z| < R2}. Prove that D1 and D2 are conformally equivalent.
1
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1answer
30 views

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$

Show that for all $z \in \overline{D}(0;1)$, $(3-e)|z| \leq |e^z - 1|\leq |z|(e-1)$ I think I'm supposed to use the following chain of inequalities $$|e^z -1|\leq e^{|z|}-1 \leq |z|e^{|z|}$$ But ...
0
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1answer
32 views

bounded components of the intersection of two planar domains

It seems to be intuitively clear that if U is a domain in the plane having a bounded complementary component C, then C is also a complementary component of the intersection of U with an open disk D ...
1
vote
1answer
76 views

Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
1
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1answer
60 views

Help on line integral $\int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$

I need help on the following line integral: $$\int\limits_\gamma f dz = \int\limits_\gamma \frac{1}{g} dz = \int\limits_\gamma \frac{1}{(z + 1)(z + 2) \cdot \ldots \cdot (z + r)} dz$$ For a fix $r \in ...
2
votes
1answer
65 views

Are derivatives actually bounded?

Suppose you a function $f$ which is differentiable, with the property that $$ f^{(n)} (0) = (n!)^2 $$ And in general $$ f^{(n)} (a) = O((n!)^2)$$ For any $a \in \mathbb{R}$. This function ...
1
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1answer
38 views

Prove that $u\leq v$ everywhere.

Let $u$ be a subharmonic function on an open set $U$ in $\mathbb{C}$, and let $v$ be an upper semicontinuous function on $U$ such that $u\leq v$ almost everywhere. Prove that $u\leq v$ everywhere. ...
1
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1answer
29 views

complex modulus and square root

I am failing to understand something about complex square roots: If we fix the argument $\theta\in(0,2\pi],$ that is we take the positive real line as branch cut, than for $z=r\mathrm{e}^{i\theta}$, $...
3
votes
2answers
38 views

Prove or refute that $\{p^{1/p}\}_{p\text{ prime}}$ to be equidistributed in $\mathbb{R}/\mathbb{Z}$

I've tried follow the Example 3 (see minute 30'40" of the reference), where is required the related Theorem (stated at minute 21') combined with Serre's formalism for $\mathbb{R}/\mathbb{Z}$ (also ...
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2answers
42 views

Complex Frequency Shifting in Fourier Transform

When dealing with Fourier transforms, it is often useful to take advantage of the following property in order to simplify work: $$\mathcal{F}(e^{i\omega_0t}f(t))=G(\omega-\omega_0)$$ where $G(\omega)...
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0answers
29 views

The complex version of the Riemann-Lebesgue lemma

I can't prove the complex version of the Riemann-Lebesgue lemma. $$ f(x) \in \mathbf{C} \\ a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \cos nx \, dx, \quad b_n = \frac{1}{\pi} \int_{-\pi}^{\...
0
votes
1answer
31 views

An application of the open mapping theorem

Let $U\subseteq \mathbb C$ be a domain and $a,b,c \in \mathbb R$ with $a^2+b^2>0$. Determine all on $U$ holomorphic functions $f$ which satisfy: $a\cdot Re(f) + b\cdot Im(f) +c = 0$. I ...
0
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0answers
35 views

$(B_t)_{t\ge 0}$ be Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t}$ is an analytic function. [on hold]

Let $(B_t)_{t\ge 0}$ be a one-dimensional Brownian motion. Then $\xi \mapsto \mathbb E e^{i\xi B_t} \; \text{for all} \; t\ge 0, \xi \in \mathbb{R}$ is an analytic function. A more general question ...
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2answers
22 views

Residue point lies on curve

The residue theorem states: Suppose $U$ is a simply connected open subset of the complex plane, and $a_1,\ldots,a_n$ are finitely many points of $U$ and $f$ is a function which is defined and ...
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0answers
24 views

Limit of nonnegative analytic functions

Let $\{g_n\}$ be a sequence of analytic functions on the closed unit disk $\overline{D_1}$ that converges uniformly to $g$ such that $g_n$ is never zero in the open unit disk $D_1$ for all $n$, then $...
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1answer
38 views

Laurent expansion - Faster technique

I'm currently preparing for an exam in complex analysis. There is a type of exercise, where I need to compute Laurent expansions about different places. However, my ...
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3answers
65 views

If $|z|=\sqrt{a^2+b^2}$, then what is $z$?

Perhaps I’m having some difficulty understanding the complex plane. Say you have a complex number $z=a+bi$, where $a$ is the real part and $b$ is the imaginary part. Why do you plot the real part on ...
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0answers
20 views

Composition operators and analytic function theory [closed]

Let C(U) denote the collection of complex valued continuous functions on U. For à ,a continuous self-map of U, we denote Cà on C(U) in the obvious way. Determine those Ã's for which Cà is one-to-one.
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1answer
38 views

How Do We Know How To Expand Function As Laurent Series

I'm a little confused about a one part of a specific example of a Laurent series that was given by a prof. It seems like it should be pretty straightforward. I want to expand $f(z)=\frac{1}{z^2-z}$ as ...
0
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0answers
32 views

To solve the radius of convergence of a power series in complex analysis [duplicate]

Let $f(z)=z+\sum_{n=2}^\infty a_nz^n$ have a positive radius of convergence. Does there exist a series $g(z)=z+\sum_{n=2}^\infty b_nz^n$, satisfying $$ f(g(z))=z\text{?}\tag{49} $$ Does this ...
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1answer
38 views

Power series expansion of a complex function problem

I don't know what is the function of the "sup" in $\lim \sup_{n\to \infty} |\beta_n|^{1/n}$ and how to Compute the first three terms of the Laurent expansion of $1/f (z)$ about $z = 0$
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1answer
32 views

homeomorphisms of the real line

Given a homeomorphism $h$ of the extended real line. Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$? Any hints are ...
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votes
0answers
64 views

It is possible to use the Zeta Function as primality test? [closed]

It is possible to use the Zeta Function as primality test? $$\displaystyle\sum_{n=1}^\infty\dfrac1{n^s} = 1+\frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+ ... $$ Where can I find the non-trivial zeros ...
3
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2answers
48 views

Prove that $\overline{f(z)}$ is differentiable at $a \in D(0;1)$ if and only if $f'(a)=0$

Let $f$ be holomorphic in $D(0;1)$ and define $k$ by $k(z)=\overline{f(z)}$. Prove that $k$ is differentiable at $a\in D(0;1)$ if and only if $f'(a)=0$. What I tried was first, assuming $k$ is ...
26
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2answers
368 views

If $f$ is a smooth real valued function on real line such that $f'(0)=1$ and $|f^{(n)} (x)|$ is uniformly bounded by $1$ , then $f(x)=\sin x$?

Let $f : \mathbb R \to \mathbb R$ be a smooth ( infinitely differentiable everywhere ) function such that $f '(0)=1$ and $|f^{(n)} (x)| \le 1 , \forall x \in \mathbb R , \forall n \ge 0$ ( as usual ...
2
votes
1answer
108 views

Compute the sum of the series.

I just see the equality in my textbook, but I really have no idea how it arises (maybe it is obvious to the author), and it seems Fourier methods are not applicable. I would appreciate if someone ...
3
votes
1answer
35 views

Calculating the radius of convergence for Taylor series

$$g(z)=\dfrac {\sin z-z+\dfrac{z^{3}}{6}}{\cos z-1}$$ What is the radius of convergence of the Taylor series of $g$ centered at 0. My thought was to use the Cauchy-Hadamard formula to calculate it. ...
1
vote
1answer
35 views

A problem concerning characterization entire function

Let $f$ be holomorphic on $C$ and suppose $P$ is a polynomial in $z$,so that for some constant $M$ one has $$|f(z)|\leq M |P(z)|$$ for all $z\in C$. Show that there is a constant $C$ so that $f(z)=CP(...
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1answer
66 views

A question about Maximum Principle in complex analysis

Let $f$ be holomorphic on open set containing $\overline{D}$,prove that there exists $z_{0}\in \partial D$, such that $|\dfrac{1}{z_{0}}-f\left(z_{0}\right)|\geq 1$ This problem can be solved by ...
4
votes
2answers
40 views

Winding number of a polynomial

Consider $f(z) = c_n z^n + ... + c_1 z + c_0$, where $c_n\ne 0$. Let $C_R$ be the circle of radius $R$ centred at the origin, oriented counterclockwise. Prove that the winding number of $f\circ C_R =n ...
0
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0answers
25 views

Uniformity of Analytic Function Growth Order

Let function $f(z),\,z=re^{i\theta}\in\mathbb C$ for $r,\theta\in\mathbb R$ on the complex plane $\mathbb C$, be analytic in the interior and continuous on the whole closed domain of the wedge $|\...
0
votes
1answer
61 views

Why is the winding number defined as $\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}dz$?

I'd appreciate some clear explanation as to why the number is defined as such. I think in my book, in the proof of the argument principle, it seems like this integral pops out of the blue, without ...
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0answers
34 views

Why $\lim_{z\to 0}\frac{(\bar{z})^2}{z^2}$ doesn't exist? [duplicate]

Why this limit does not exist? $$\lim_{z\to 0}\frac{(\bar{z})^2}{z^2}$$ I know that the limit exists and equals to $1$ is $z$ is real. When $z$ is complex, I tried $$\frac{(\bar{z})^2}{z^2}=\frac{e^...
0
votes
0answers
36 views

Proving that an infinite series equal a finite series

Suppose we have a function $f(z)$, which has $m$ isolated singularities, which are non-integers (say, $z_1$, $z_2$,...,$z_m$). Define $H(z):=\frac{\pi f(z)}{\sin(\pi z)}$. Assume that there exists a ...
0
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0answers
18 views

Residue at all integers of complex function involving sines

Given $H(z)=\frac{\pi f(z)}{\sin(\pi z)}$, where $f(z)$ is some function which has isolated singularities only at non-integers, is it correct to calculate its residue at $z=k$ (for $k\in\mathbb{Z}$) ...
3
votes
2answers
48 views

Analytic continuation of $\sum (z/a)^n$

I'm having trouble continuing this function beyond its convergence radius, $R=a$. $$f(z)=\sum (z/a)^n$$ Given the context (a textbook in complex analysis) I suspect it should have a simple closed-...
4
votes
1answer
73 views

(Non-)Canonicity of using zeta function to assign values to divergent series

This article http://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/ got me thinking about the "identity" $$1 + 2 + 3 + \cdots = -1/12,$$ and I wanted to convince myself there ...
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0answers
21 views

Find the image of the set U={z∈C∣-π/2<Re z<π/2} under the function f(z)=sinz. [closed]

How to Find the image of the set U={z∈C∣-π/2 1.What is the image of the line segment L1=(-π/2,π/2) (on the real axis) under f? 2.What is the image of the imaginary axis L2={iy∣y∈R} under f? 3.What is ...
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0answers
21 views

Using 2D Parseval-Plancheler theorem to solve an equation

In the context of a digital communications problem I have to solve the following equation with respect to $\tilde{\tau}$: \begin{eqnarray*} &&Im\Big\{\Big(\int\limits_0^{T_0}r^{*}(t)g(t-\tilde{...
5
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0answers
72 views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and ...
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2answers
31 views

Zeros and poles of some meromorphic 1-forms on the riemann sphere

Let $X=\mathbb C_{\infty}$ be the Riemann sphere with the local coordinates $\{z\ ,1/z\}$. I want to show the following two statements: i) There does not exist any non-vanishing holomorphic 1-form on ...
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0answers
19 views

Proving non-existence of a Möbius transformation with cross-ratio

I have difficulties in understanding the following solution: Is there a Möbius transformation mapping the unit circle on the unit circle and the circle around $0$ with radius $1/2$ on the circle ...
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0answers
24 views

A non-constant holomorphic map $F$ between riemann-surfaces is an isomorphism

I want to show the following: Let $F:X\rightarrow Y$ be a non-constant and holomorphic map between compact riemann surfaces with $genus(X)=genus(Y)\geq 2$. In the above it holds that $F$ is an ...
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1answer
25 views

All non-zero entire functions in exponential form: still problematic?

I saw the question $f$ is entire without any zeros then there is an entire function $g$ such that $f=e^g$ I understand that for any non-vanishing entire function $f(z)$: If there exists an entire ...
3
votes
1answer
44 views

$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$, for complex variable $z$.

I want to find this limit for complex variable $z$ $$\lim_{z\to -1} (z+1) \sin(\frac{1}{z+1})$$ In the real case I know $\sin(z)$ is bounded by $-1, 1,$ and the limit is $0$. But in the complex case ...
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2answers
24 views

Linear Integral in complex plane

Integrate using Indefinite Integration and Substitution of Limits Integration symbol with $c$ in denominator $ \mathrm{Re}(z) \, \mathrm{d}z $, $C$ the shortest path from $1 + i$ to $5 +5i$.
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0answers
24 views

calculating curvilinear integral by residue theorem

Calculate the following integral by transposing to a curve integral and then using the residue theorem: $\displaystyle \int_{0}^{2\pi}{\frac{e^{int}}{C-e^{it}}dt}, \qquad |C|\ne1, n\in \mathbb N$. ...
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votes
2answers
79 views

Prove $ \int_{-\infty}^{\infty} e^{-\alpha(x-c)^2} x^{2n} dx =\int_{-\infty}^{\infty} e^{-\alpha x^2} x^{2n} dx$

I understand $$ \int_{-\infty}^{\infty} e^{-\alpha x^2} x^{2n} dx =\frac{(2n-1)!!}{2^n}\sqrt{\frac{\pi}{\alpha^{2n+1}}} \quad (\alpha \in \mathbb{C}, \, \text{Re} \,\alpha>0) $$ But I can't prove ...
0
votes
2answers
28 views

How is $\left| \exp(iaRe^{i\theta}) \right|\le e^{-aR\sin\theta}$?

In one book on complex variables, in the proof of Jordan's Lemma, For any constant $a>0$, and any radius $R>0$, it is stated that $\left| \exp(iaRe^{i\theta}) \right|\le e^{-aR\sin\theta}$. I ...
8
votes
2answers
490 views

$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$ must be zero and it isn't

I'm trying to evaluate the integral $$\int_{-\infty}^\infty \frac{\sin (t) \, dt}{t^4+1}$$ using residue and complex plane integration theory. Let $f(t):=\frac{\sin (t)}{t^4+1}$, $f(z):= \frac{\sin (...