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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Examples of Weil's explicit formula

In Bombieri, PROBLEMS OF THE MILLENNIUM: THE RIEMANN HYPOTHESIS, Clay Mathematics Institute (2000), from page 8, V. Further evidence: the explicit formula the author tell us that there is a ...
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31 views

Existence of such a meromorphic function?

Is there a function $f$ that is holomorphic on $\mathbb{C}-\mathbb{Z} $ and maps into or onto $\mathbb{C}-\mathbb{R}$ ? Into or onto $\mathbb{C}-\mathbb{R}^{+}\cup\{ {0} \}$? All I have been able to ...
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11 views

How does the author derive this (Difference of analytic functions evaluated at two points)

The conditions are $f:U\to V$ is holomorphic and injective. I basically have 2 questions: Q1) How did the author get $f(z)-f(z_0)=a(z-z_0)^k+G(z)$? Q2) What does "vanishing to order $k+1$" mean? ...
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85 views

On real part of the complex number $(1+i)z^2$

Find the set of points belonging to the coordinate plane $xy$, for which the real part of the complex number $(1+i)z^2$ is positive. My solution:- Lets start with letting $z=r\cdot e^{i\theta}$. ...
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18 views

find the principal part of the function at its isolated singular point

find the principal part of the function at its isolated singular point, and determine whether that point is a pole, a removable singular point, or an essential singular point. (a) f(z) = z exp(1/z) (...
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23 views

The difference between Taylor and Laurent expansions for Holomorphic functions

I have encountered 2 similar but different theorems on expansions of holomorphic functions to power series, but am not sure how exactly do they differ. Is correct that any $f: U \rightarrow \mathbb{C}...
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69 views

Circles in complex plane.

Find the real value of a for which there is at least one complex number satisfying $|z+4i|=\sqrt{a^2-12a+28}$ and $|z-4\sqrt{3}|\lt a$. My solutions:- Graphical solution:- $|z+4i|=\sqrt{a^2-...
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49 views

Why is holomorphic function with non-zero derivative a conformal map?

I am new to complex analysis, interested to know why non-zero derivative implies a conformal map. Intuitively, I would think that non-zero derivative means the function is non-constant. Why would ...
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56 views

Can I show that $\int_{\gamma(0;r)} \frac{1}{z-a} dz = 0$ when $|a|>r>0$ without using Cauchy Theorem?

I encountered this problem as a previous result of an exercise in a text book way before proving Cauchy Theorem, so I think there must be another way to prove it without it. Show that $\int_{\...
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51 views

Show that for $f$ analytic in $B(0,2)$, $\max_{|z|=1}|\frac{1}{z}-f(z)|\ge 1$?

Let $f:B(0,2)\to \Bbb C$ be an analytic function. Show that $$\max_{|z|=1}\left|\frac{1}{z}-f(z)\right|\ge 1.$$ I tried to write $f(z)$ as power series since it is analytic, it doesn't seem work. I ...
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83 views

Prove $f$ is identically zero in $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} (z)| < 1,|{\mathop{\rm Im}\nolimits} (z)| < 1\} $

Let $\Omega = \{ z \in \Bbb C:|{\mathop{\rm Re}\nolimits} z| < 1,|{\mathop{\rm Im}\nolimits} z| < 1\} $ and consider the function $f:\bar\Omega\to\Bbb C$ continuous on $\bar\Omega$, analytic in ...
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14 views

Finding domain where this complex logarithm identity holds

I have not had much exposure to more advanced complex analysis exercises, although this one might be easy, and the solution just completely escapes me. I am determining the set of complex numbers for ...
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34 views

Analytic continuation of $\sum_{n=0}^{\infty} e^{-x E_n}$

Suppose we define a function $f(x)$ by the following sum: $$f(x)= \sum_{n=0}^{\infty} e^{-x E_n}$$ where $E_n$ is a sequence which is at most $O(n)$. It is known $f(x)$ does not form a natural boundry ...
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25 views

Local normal convergence equivalent to compact normal convergence

Let $X$ be an open subset of $\mathbb{R}^m$ and let $f_n\colon X\to \mathbb{C}$ be complex-valued functions. Then one has the following two notions: $\textbf{1.}$ The series $\sum\limits_{n=0}^{\...
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52 views

Cauchy's Integral Formula implies Holomorphicity?

Is the converse direction of the Cauchy Integral Formula true? Meaning, if $f:\mathbb{C}\supseteq U\rightarrow\mathbb{C}$, and $$\forall a \in U \space \space f(a) = \frac{1}{2\pi i}\int_{|z-a|=R}\...
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41 views

Analytic curve divides disk into two Jordan regions

Let $\gamma:(0,1)\rightarrow\mathbb{C}$ be an analytic Jordan arc. It seems natural to me that for every $\gamma(t_0)$ we can find a disk with center $\gamma(t_0)$ that is divided by $\gamma$ into two ...
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28 views

Compact convergence of polynomials

I Want to prove that there is no Sequence of complex polynomial that converges to $f(z)=\frac{1}{z} $ on $D=\mathbb{C} \setminus \{0\}$. Suppose there is a Sequence of complex polynomial converging ...
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1answer
64 views

Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
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32 views

Continuity Of Complex Valued Functions

Consider the following complex valued function: $$f(z)=(2+z)Arg(z)$$ Does $f(z)$ have removable discontinuities? (Note: $Arg(z)$ denotes the principal argument.) The following is my approach: We ...
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52 views

Geometric Description Of a Set In The Complex Plane

$$S_1=\left\{z:Im\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ $$S_2=\left\{z:Re\left(\frac{z-z_1}{z-z_2}\right)=0, z_1,z_2 \in \Bbb C\right\}$$ Can someone help me with the ...
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31 views

A property about the automorphisms of $\mathbb{D}$

I want to prove the next proposition: if $T$ is a Möbius transformation from $\mathbb{D}$ to $\mathbb{D}$, then \begin{equation} \left|\frac{T(z_1) - T(z_2)}{1 - \overline{T(z_1)} T(z_2)}\right| = \...
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14 views

Argument principle for meromorphic forms on Riemann surfaces

Let $X$ be a compact Riemann surface and $D \subset X$ a compact domain with boundary $\partial D$. Let $\omega$ be a meromorphic $1$-form in a neighborhood of $D$ which does not have neither zeros, ...
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34 views

Determining if a function is harmonic in a fast way

Determine which function is harmonic in $\mathbb R^2$: $$\text{a) } y^2 \qquad \text{b) }x^2 + y^2\qquad \text{c) } e^x\qquad \text{d) }\operatorname{Im}((x + iy)^5)$$ I had this question come up ...
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30 views

On the existence of a Möbius transformation

Consider a rational function $f(z)\in\mathbb{R}(z)$ with no poles/zeros on the unit circle $\mathbb{T}=\{z\in\mathbb{C}\,:\, |z|=1\}$. Does there always exist a Möbius transformation $$ \rho\colon \...
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25 views

Show $|f(z)|<3$ for each $z\in B(0,1)$?

If $f:B(0,2)\to\Bbb C$ be an analytic function satisfying $|f(z)-2|<1$ for each $z\in \Bbb C$ such that $|z|=1$. show that (a) $|f(z)|<3$ for each $z\in B(0,1)$? (b) $f(z)\neq0$ for each $z\...
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35 views

Existence of biholomorphic map from unit disk to itself that interpolates one set of points

How do you prove that given two points $z_{1}, w_{1} \in D = \{z: |z|<1\} $, there exists a biholomorphic (bijective and analytic) function $f: D \to D$ such that $f(z_{1}) = w_{1}$? Perhaps using ...
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How does one prove that two punctured disks are conformally equivalent? [closed]

Let D1 = {z: 0 < |z| < R1} and let D2 = {z: 0 <|z| < R2}. Prove that D1 and D2 are conformally equivalent.
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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 ...
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38 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 ...
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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 ...
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62 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 ...
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69 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 ...
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1answer
39 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. ...
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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 $z=r\mathrm{e}^{i\theta}$, $...
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39 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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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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30 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}^{\...
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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 ...
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$(B_t)_{t\ge 0}$ 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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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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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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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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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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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 ...
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43 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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51 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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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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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 ...
27
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399 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 ...
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1answer
109 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 ...