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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2
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2answers
86 views

closed path, winding number, Jordan contour

If $ D$ is a domain in $\Bbb C$, $z_0\in \Bbb C\setminus D$, and $\gamma$ is a closed, piecewise smooth path in $ D$ for which the winding number $n(\gamma, z_0)\ne0$, show that there is a Jordan ...
1
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0answers
43 views

An analytic function is the identity function

Let $\Omega$ be a bounded domain and $\phi:\Omega \rightarrow \Omega$ a conformal mapping. Let $P\in \Omega$ be such that $\phi(P)=\phi'(P)=1$. Show that $\phi$ must be identity.
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0answers
45 views

$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R ...
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0answers
28 views

Biholomorphic map to a belt. Why?

The exercise is this: We have a set A = {z | |z-i|>1 and Im(z)>0} We want to find a biholomorphic map f that would map that area A into a belt. So the solution is apparently f(z) = 1/z which maps it ...
2
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1answer
27 views

Does there exist a conformal $\phi: D\rightarrow\Omega\cup\{\infty\}$?

Let $\gamma$ be a Jordan curve and $\Omega$ the unbounded connected component of $\mathbb{C}\setminus\gamma$. $\Omega$ is not simply connected in $\mathbb{C}$, but $\Omega\cup\{\infty\}$ is simply ...
-1
votes
1answer
26 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
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1answer
40 views

“Direct” derivation of exponential form of the Riemann zeta function.

There is the identity $$ \zeta(s) = \exp\left(\sum_{n=2}\frac{\Lambda(n)}{\log(n)} n^{-s} \right) $$ for $\Re(s)>1$. Apparently there are quite a few possibilities to derive this. I am out to "try ...
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0answers
46 views

Claim about holomorphic extension

Prove or disprove the following claim. "For all continuous $f : S(0, 1) \to R$, there is a holomorphic $g : B(0, 1) \to C$ which extends to a continuous ${h : \overline {B(0, 1)}} \to C$ such ...
0
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1answer
47 views

Conformal map is an isometry

I have the upper half-plane $\mathbb H$ with the metric given by $$\mathrm ds^2=\frac{1}{y^2} (\mathrm dx^2+\mathrm dy^2)$$ and the unit disk $\mathbb D$ with the metric given by $$\mathrm ...
1
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1answer
24 views

Resolvent $R(\lambda,A)x \to 0$ as $|\lambda| \to \infty$

If I have a closed operator $A:D(A) \to X$, not necessarily bounded on a Banach space $X$, and the resolvent is unbounded, can I show for a fixed $x \in X$ that $$R(\lambda,A)x \to 0$$ as ...
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3answers
60 views

Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
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1answer
39 views

On a simple application of Paley-Wiener theorem and related doubts

Let $$F(x)=\frac{ \left\{ x \right\} }{e^{\sqrt{x}}},$$ be supported on $ \left( 0,\infty \right) $, where $ \left\{ x \right\} $ is the fractional part function. Then $F\in L^2(0,\infty)$ and the ...
4
votes
0answers
81 views

Is right this application of Hadamard three-lines theorem for $ \frac{\zeta(s)}{s}- \frac{d\zeta(s)}{d\sigma}$?

Let the complex variable $s=\sigma+it$, then from the following identity valid for $\sigma=\Re s>1$ $$\zeta(s)=s\int_1^\infty \frac{[x]}{x^{s+1}}dx$$ where $\zeta(s)$ is the Riemann Zeta function, ...
0
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0answers
33 views

An Alternate proof of Nyquist-Shannon

This problem is from Basic Complex Analysis, Part 2A, by Barry Simon. This problem will provide an alternate proof of the strong from of the Nyquist-Shannon sampling theorem (Theorem 6.6.16 of Part ...
13
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2answers
137 views

Egg vs. chicken: trig functions, exponential, real and complex

This is something I was shaky about when I took calculus, real analysis, and then complex analysis. Specifically, is the following chain of definitions circular in any way? Define the set ...
1
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1answer
33 views

Laurent series expansion of $\frac{z^2-1}{z^2+1}$

Given $f(z) =\frac{z^2-1}{z^2+1}$, I need to find it's Laurent series expansion at open disk $\sqrt{2} < |z-1| < +\infty$ So at first I've found that, at $z=\pm i$ function is not ...
3
votes
1answer
67 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
3
votes
0answers
48 views

Use Residue Theorem to Sum Series

Show that $$\sum_{n=- \infty}^{\infty} \frac{1}{(3n-1)^2} = \frac{4 \pi^2}{27}$$ I'm pretty sure I need to use the Residue Theorem to sum the series, but I'm unsure how to apply it. Here's what ...
2
votes
2answers
55 views

Rouche's Theorem application for $z^6-5z^4+3z^2-1$ in $|z|\leq 1$

Find the number of roots of $f(z)=z^6-5z^4+3z^2-1$ in $|z|\leq 1$ Taking $g(z)=1$ would be the obvious choice, but it's not the right one. The next choice would be $z^6-1$ because we know the roots ...
2
votes
0answers
30 views

Determine value of series using Euler's identity and geometric series

Use Euler’s Identity and the geometric series to determine the values of the series $\sum_{k=0}^\infty r^k \cos(k\theta)$ and $\sum_{k=0}^\infty r^k \sin(k\theta)$, where $|r| < 1$ and $\theta \in ...
0
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0answers
16 views

Proving that a function must be constant using Complex Analysis [duplicate]

Fix nonzero ω1,ω2 ∈R. Suppose that f is an entire function which satisfies $$f(z + ω1) = f(z + iω2) = f(z)$$ for all $z ∈\Bbb C$. Prove that f must be constant. My first and immediate thought upon ...
2
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0answers
51 views

If for every $z\in G$ there is $n$ such that $f^{(n)}(z)=0$, then $f$ is a polynomial. [duplicate]

Let $G$ be an open and connected set of the complex plane $\mathbb{C}$ and $f:G\rightarrow \mathbb{C}$ be analytic, such that for each $z\in G$ there is $n=n(z)\in \mathbb{N}$ such ...
2
votes
1answer
42 views

Computing a summation using Maclaurin series and infinite products

Using the Maclaurin series for $\sin z$ and $\sinh z$, as well as the infinite products $$\sin z = z\prod_{n=1}^\infty\left(1 - \frac{z^2}{n^2\pi^2}\right)$$ and $$\sinh z = ...
0
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3answers
55 views

The existence of the derivative of a function at a point implies the continuity of the function at that point.

In preparation for my first course in complex variables I am trying to read Brown and Churchill, Complex Variables, 8th edition. On page 59 they give the following justification for the statement in ...
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0answers
58 views

Radius of convergence and expansion [closed]

"Assume that a complex function $f(z)$ is regular in a neighborhood of $z = 0$ and satises $$f(z)e^{f(z)}= z$$ Write the polynomial expansion of $f(z)$ at $z = 0$ and find its radius of convergence." ...
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0answers
49 views

Weierstrass theorem and the series $\sum\frac{z^k}{k^2}$ and its derivative

The Weierstrass theorem states that if a series with analytic terms, $f(z)=f_{1}(z)+f_{2}(z)+\cdots +f_{k}(z)+\cdots $ converges uniformly on every compact subset of a region $\Omega$, then the sum ...
0
votes
1answer
31 views

Let $p(z)$ be a polynomial, $\, z \in \mathbb{C}$. Show that if $\left|p(z) \right| \leqslant \left|e^z \right|$ for every $z$, then $p=0$.

Let $p(z)$ be a polynomial, $\, z \in \mathbb{C}$. Show that if $\left|p(z) \right| \leqslant \left|e^z \right|$ for every $z$, then $p=0$. I proved that if $f$ is entire and $\left|f(z) \right| ...
0
votes
1answer
24 views

Prove that if the image $f \left( \Omega \right)$ is compact in the $w$-plane, then it must be a single point

Let $f:D \to \mathbb{C}$ be analytic on the domain $D$ (open and connected). Prove that if the image $f \left( \Omega \right)$ is compact in the $w$-plane, then it must be a single point; that is $f$ ...
1
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2answers
46 views

Computing an infinite sum using the Residue theorem

Show that $$\sum_{n=-\infty}^\infty \frac{1}{(3n-1)^2} = \frac{4\pi^2}{27}$$ Here is what I tried so far. I know that I can use the Residue theorem to solve a summation of this form. ...
1
vote
1answer
32 views

Prove that $\left|f^{(n)} \left(z_0 \right) \right| \leqslant \frac{n! M}{R^n}$ for every $n$.

Let $f$ be analytic on a domain $\Omega$ containing the closed disk of radius $R$ centered at $z_0$. Show that if $\left|z-z_0 \right| = R \implies \left| f(z) \right| \leqslant M$ then $\left|f^{(n)} ...
0
votes
1answer
47 views

What is the principal value of this number?

Find all values of the logarithm of each of the following numbers and state the principal value. Put answers in the form a+ib. $e^{i \pi /3}$ I have that it is equal to $\frac{\sqrt3}{2}+\frac12 ...
2
votes
2answers
22 views

Show that $f(z)=\frac{1}{z^n}$ for all $z \in U$

Let $U=\{z \in \Bbb{C}:0<|z|<1\}$,$n \in \Bbb{N}$. Show that if $f$ is holomorphic in $U$, $f(\frac{1}{5})=5^n$ and $|z^nf(z)|\geq 1$ for all $z\in U$, then $f(z)=\frac{1}{z^n}$, $\forall ...
0
votes
1answer
8 views

Radius of convergence of complex power series using Cauchy's integral formula

I have a question as follows. Let $$f(z)=\frac{\sin z}{(z-1-i)^2}$$ and $$a_n=\frac{f^{(n)}(0)}{n!}$$ Determine the radius of convergence of $$\sum_{n=0}^{\infty}a_nz^n$$ In my class we have ...
0
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0answers
45 views

How to take the derivative of $ i^z$?

I know that to take the derivative of f(t)= $a^t$ f'(t) =$ ln(a) * a^t * t'$ So to take derivative of $i^z$ We do $ln(i) * i^z * z'$ But I need help on the next step Thank you
0
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1answer
23 views

Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
0
votes
1answer
23 views

Function analytic on annulus bounded by $|z|^2$

This problem comes from an old prelim "Let $f$ be analytic on an open neighborhood of the annulus $1\le |z|\le 2$. Assume that $|f(z)|\le 1$ when $|z|=1$ and $|f(z)|\le 4$ whenever $z=2$. Show that ...
0
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1answer
17 views

Taking second partial derivatives of a function composition

Let $H(u,v)$, $\, u:=u(x,y)$, and $\, v:=v(x,y)$ be real valued functions. I am having trouble taking the second partial derivatives $H_{xx}$ and $H_{yy}$ of this function composition using the chain ...
0
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2answers
44 views

Is the complex function $f(z) = Re(z)$ differentiable?

I am preparing to take my first course in complex variables. I am reading some lecture notes online. They claim that the function $f(z) = Re(z)$ is continuous but NOT differentiable. I know the ...
1
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1answer
36 views

complex analysis, about the identity theorem

Two functions $f$ and $g$ are analytic and never zero in the open unit disc $U$. In addition they satisfy $\frac{f'}{f} ( \frac{1}{n} )=\frac{g'}{g} ( \frac{1}{n} )$. $n=5,6,7,\dots$, find a precise ...
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1answer
36 views

Harmonic forms and functions on compact manifolds

I hope my question is not stupid, I'm studying chapter 5.1 of Claire Voisin's book "Hodge theory and complex geometry". Let $X$ be a compact manifold and $A^k(X)$ be the space of $C^\infty$ forms on ...
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2answers
42 views

Showing automorphisms on $\mathbb{C}(x)$

Let $\mathbb{C}(x)$ denote the field of rational functions over $\mathbb{C}$, the field of complex numbers. Consider the six mappings $\phi : \mathbb{C}(x) → \mathbb{C}(x)$ defined by $\phi_{1}:f(x) ...
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1answer
40 views

Area of circle in terms of Gaussian curvature

I am asking about a formula in section 2 of these notes. Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as ...
4
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1answer
62 views

Is there a closed loop in the complex plane such that for any given integer $x$, I can find a point inside the loop that has winding number $x$?

We've been discussing winding numbers in my complex course, and also Alexander polynomials and other invariants on knots in my Alg. Top. course, and the question came to me about the possibility of ...
0
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2answers
49 views

Construction of function holomorphic in $\mathbb{C}\setminus\{0, 1\}$ satisfying specific conditions

I'm learning about complex analysis, specifically (Laurent) series and residues, and need help with the following problem: Construct a function $f(z)$ holomorphic in $\mathbb{C}\setminus\{0, 1\}$ ...
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0answers
43 views

Show that $T(z)= \dfrac{az+b}{cz+d}$ is a linear map. [duplicate]

Section 96 Chapter 8 Complex Analysis by Churchill writes (without proo) that $T(z)= \dfrac{az+b}{cz+d}$ is a linear map. To my knowledge, a map is linear when $T(\alpha_1 z_1+ \alpha_2 z_2) = ...
3
votes
0answers
48 views

Let $I$ ideal of $\mathcal{O}(D)$ then $fg\in I\Rightarrow f(z)g(z)(z-z_0)^{-k}\in I$

Let us consider the ring $\mathcal{O}(D)$ of holomorphic functions on the open subset $D\subseteq\Bbb C$. Let then consider an ideal $I\unlhd \mathcal{O}(D)$ s.t. $\bigcap_{f\in ...
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0answers
9 views

Inverse trignometric functions in complex plane

I've been trying to find out where is the trignomatric function $Arccos(z)$ holomorphic in the complex plane! Now knowing that $Arccos(z)$=$(1/i)Log( z + \sqrt{ z^2 -1} )$ with $Log$ being the ...
0
votes
0answers
20 views

Laurent series of logarithm

Lets have a function $$f(z)=\ln(\frac{z-a}{z-b})$$ on the region where it is holomorphic(off course). I want to find the laurent series for this function. Now finding the taylor expansion of this ...
1
vote
1answer
21 views

Order of a meromorphic function at $\infty$

Let $f$ be meromorphic on the whole extended complex plane $\overline{\Bbb C}$. Now my teacher wrote $$ \text{ord}_{\infty}f=-\frac1{2\pi i}\int_{|z|=R}\frac{f'(\zeta)}{f(\zeta)}\,d\zeta $$ and I ...
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0answers
60 views

Let $D=\{z \in \mathbb{C}: |z|<1\}$, then which of the following are true? [duplicate]

Let $D=\{z \in \mathbb{C}: |z|<1\}$, then which of the following are true? There exists a holomorphic function $f:D\to D$ with $f(0)=0$ and $f'(0)=2$. There exists a holomorphic function $f:D\to ...