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
31 views

Investigating the convergence of a complex series along its boundary of convergence.

I can easily see why the complex series $$\sum_{n=1}^\infty nz^n$$ has a radius of convergence of $1$. My professor claims that the series diverges everywhere along the boundary, i.e., when $z$ ...
0
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3answers
60 views

Number of complex n'th roots of unity

I just started to read Freitag/Busam's book on complex analysis. They prove that there are exactly $n$ complex $n$th roots of unity (page 9). But I only understand the existence proof. In it they ...
2
votes
7answers
118 views

Prove or disprove: $|\sin z|<1$ for all $z \in \mathbb C$

Prove or disprove: $|\sin z|<1$ for all $z \in \mathbb C$ I took some good advice and try $z=iy$ for $y\in \mathbb R$, so $$\sin z= \sin iy = \frac{e^{i(iy)}-e^{-i(iy)}}{2i}$$ ...
7
votes
3answers
126 views

Find the error in following reason $(-z)^2=z^2 \implies \log(-z)^2=\log(z)^2 \implies2\log(-z)=2\log(z)\implies \log(-z)=\log(z)$

Find the error in following reason \begin{align*} (-z)^2=z^2 &\implies \log(-z)^2=\log(z)^2\\ &\implies2\log(-z)=2\log(z)\\ &\implies \log(-z)=\log(z) \end{align*} I think the error is ...
1
vote
0answers
20 views

Is it true that $2\max_{|z|=1} | z^n+ (\sum_{ i=2}^{n-1} a_iz^i ) + a_1 z+a_0| \geq \max_{|z|= 1} |a_1z+a_0|$?

I come across to prove or disprove an inequality which is $$2\max_{|z|=1} | z^n+ (\sum_{ i=2}^{n-1} a_iz^i ) + a_1 z+a_0| \geq \max_{|z|=1} |a_1z+a_0|,$$ where $z\in \mathbb{C}$ and $a_i$ are real. ...
0
votes
0answers
62 views

Proof of exponential complex constant holomorphic

If we let $ g(z) = e^{f(z)} $ and $f$ is holomorphic on a region G, how do we prove that $f$ is a constant if $g$ is a constant? Do we just simply find the derivative of $g(z)$ and set it to zero, ...
1
vote
0answers
36 views

A simply connected region $D$ that contains the boundary of $S$ contains $S$

If $D\subseteq X$ is a simply connected subspace of the topological space $X$ and [add assumptions here] and $S\subseteq X$, $\partial(S)\subseteq D$, then $S\subseteq D$. It doesn't seem to be ...
1
vote
2answers
26 views

Complex argument computation

Define $argument$ of $z$ to be $[arg z] := (\theta \in \Re : z=|z|e^{i\theta})$ How do we then find $[\sqrt2^i]$ and $[i^{\sqrt2}]$ ? $[\sqrt2^i]$ = $\frac{ln2}{2}$ from other sources. But can ...
1
vote
1answer
38 views

Showing a function is analytic

How to show this function is analytic? $f(z) = e^y \sin x + ie^y \cos x$ on $C$ Note: C is the domain. Attempt Partial derivatives of one equation are equal therefore cauchy reimann thm holds
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2answers
146 views

A question in Rudin's Real and Complex Analysis

My book at hand is Walter Rudin's Real and Complex Analysis, 3ed, 1987 McGraw-Hill. In Chap8: integration on product spaces, Theorem 8.16 is stated as: Let $\mu$ be a $\sigma$-finite positive ...
2
votes
0answers
57 views

Finding the constant in Stolz region

Given two points $z_1, z_2 \in \mathbb{C}$ such that $|z_1| < 1$ and $|z_2|<1$, there exists $K > 0$ such that for any points $z \neq 1$ in the closed triangle with vertices $z_1, z_2,$ and ...
0
votes
1answer
154 views

Landau's proof of MMT (Maximum Modulus Theorem)

This is an exercise of Newman's book ans says this is a proof, due to Landau, of the maximum modulus theorem, but I'm not getting how to prove it. Suppose $f$ is an analytic inside and on a circle ...
4
votes
1answer
88 views

Biholomorphic map from a disk to its quarter

Could you tell me how to find a biholomorphic map from a unit disk $D$ to $\{ |z|<1 \ : \ \Re z >0, \ \Im z >0 \}$? I know that mapping of the form : $\frac{az+b}{cz+d}$ won't work. Also, ...
0
votes
1answer
27 views

Sufficient condition for a differentiable function constant

If $f = u+iv$ complex differentiable in a ball centered at $(0,0)$ and $au^2+bv^2$ is constant where not both $a$ and $b$ zero. Then prove that $f$ is constant. If $a=0$ or $b =0$ I have finished ...
0
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1answer
52 views

Describing Polynomials with Real Coefficients that are the Real Parts of Analytic Functions on $\mathbb{C}$

Describe those polynomials $$P=a + bx + cy + dx^2 + exy + fy^2$$ with real coefficients that are the real parts of analytic functions on $\mathbb{C}$. Idea (1): We are given that $P$ is the real part ...
1
vote
1answer
23 views

Determining order of poles.

I'm having some trouble understanding how to tell the order of poles. I understand simple cases I've seen online, but how about in something like ($z^{2}+1$)/($z^{3}-1)$? There's a pole at 1 and I ...
1
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3answers
40 views

What is the length of the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$?

Let $C$ be the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$. What is the length of $C$? Would the length of $C$ be $5$ or $10$? I think $r=5$ so I am not sure whether that would be the ...
4
votes
1answer
82 views

How to determine the radius of convergence if the Taylor series cannot be written in a neat way?

I am trying to evaluate the radius of convergence of Taylor series centered at zero of function $$f(z)=\frac{\sin(3z)}{\sin(z+\pi/6)}$$ I guess the answer should be $\pi/6$ because the function will ...
1
vote
1answer
103 views

Complex Analysis (Contour Integration)

Given complex numbers $z_1$ and $z_2$, let $[z_1, z_2]$ denote the straight line segment path from $z_1$ to $z_2$. Recall that we can parametrize this by $x(t) = z_1 + t(z_2 - z_1)$ for $t \in ...
2
votes
2answers
33 views

Proving or disproving the convergence of a sequence

I'm looking to determine the convergence of $\{e^{in}\}_{n=1}^\infty$ in $\mathbb{C}$, where $n \in \mathbb{N}$. Using the definition of convergence that iff $\exists \ \epsilon > 0: \forall \ ...
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1answer
34 views

Zeros of Complex Polynomial and Fundamental Theorem of Algebra

Let $p(z)=$ polynomial over $\mathbb{C}$. I want to show (possibly, without Fundamental Theorem of Algebra) There exists a disc $D$ in $\mathbb{C}$ such that the (possible) zeros of $p(z)$ lie ...
1
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1answer
35 views

Mapping/Area of an Image

In $\mathbb{C}$, consider the neighborhood $O := \{x+yi; (x+1)^2 + (y-1)^2 < 4\}$ and the map $w = z^2$. In terms of area, what percentage of the image of $O$ consists of image points from multiple ...
1
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1answer
32 views

Complex exponentials inequality series proof disc

If $z$ is in a closed disc $\bar{D}(0;1)$, how do we prove that $(3-e)|z|\leq |e^z-1| \leq (e-1)|z|$ ? I could attempt: $(3-\sum \frac{1}{n!})|z| \leq |\sum \frac{z^n}{n!} -1| \leq (\sum ...
0
votes
1answer
39 views

Complex exponentials inequality series proof

$(3-e)|z|\leq |e^z-1| \leq (e-1)|z|$ ? I could attempt: $(3-\sum \frac{1}{n!})|z| \leq |\sum \frac{z^n}{n!} -1| \leq (\sum \frac{1}{n!}-1)|z|$, but I'm stuck...
2
votes
1answer
52 views

Complex exponential inequality series proof

We know that $|\sum a_n|\leq \sum |a_n|$ So how do we prove that $|e^z-1|\leq e^{|z|}-1\leq |z|e^{|z|}$ ? Do I find the power series first? i.e. $|e^z-1| = |\sum \frac{z^n}{n!} -1| \leq \sum ...
1
vote
1answer
118 views

finding the center of a circle given 3 points [duplicate]

Suppose we have three complex numbers $a_1,a_2,a_3$ which are non-collinear. What is the best way to find the center of the circle that contains these three points ?
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1answer
42 views

Trying to prove a sequence of functions is increasing.

Put $\delta_n = 2^{-n}$.To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t < (k+1)\delta_n$. Define $$\psi_n(t)= ...
2
votes
2answers
136 views

Calculating an integral (using methods from complex analysis) (hints only please)

From Rudin's book, we are to calculate $\int_\mathbb{R} \Big(\frac{\sin x}{x}\Big)^2 e^{itx}dx$ where $i$ is the imaginary number and $t\in\mathbb{R}$. I'm looking for a hint on how to get started. I ...
0
votes
2answers
52 views

What are some books or online resources about the Sommerfeld-Watson method for series summation?

I'm looking for resource recommendations on the Sommerfeld-Watson summation method, i.e. the use of the residue theorem to obtain expressions like $$ \tag 1 \sum_{n \in \mathbb{Z}} g(n) = - \pi ...
0
votes
2answers
89 views

Is $\frac{1}{z}$ analytic?

Suppose I am to prove: $$f(z) = \frac{1}{z}$$ is analytic everywhere. I see there is an obvious discontunuity at $z=0$, but we can apply the residue theory, which means f(z) is indeed analytic $$ ...
0
votes
2answers
42 views

Complex Analysis: Limits

I am currently working on the following problem: Let c denote a complex constant. Then use the definition of limit to show that $$\lim_{z\to\ z_0} (z^2 + c) = z_0^2 +c$$ Definition of limit: For ...
2
votes
1answer
35 views

Limit of a cotangent function.

$$\lim_{z \to 0} -\pi^2\csc^2(\pi z) = \lim_{z \to 0} \frac{-\pi^2}{\sin^2(\pi z)} $$ How to go about this?
0
votes
1answer
43 views

Residue theorem and Angle of modular function

Let $f$ be a meromorphic function on the region $Im(z)>0$, $v_p(f)$ be the order of $p$. (The number $n$ such that $\frac{f(z)}{(z-p)^n}$ is holomorphic and non-zero at $p$.) Moreover, assume $f$ ...
2
votes
1answer
100 views

Prove that $f$ analytic, $f(x) \in \mathbb{R}$ for all $x \in \mathbb{R}$ implies $f(\overline{z})=\overline{f(z)}$

Let $U\subset \mathbb{C}$ be a nonempty connected open set such that for every $z\in U$, $\overline z\in U$. Let $f$ be analytic on $U$. Suppose $f(x)\in\mathbb R$ for every $x\in U\cap\mathbb R$. ...
2
votes
1answer
171 views

If $f$ is analytic, prove that $\overline{f(\overline{z})}$ is also analytic

Let $f$ be an analytic function in an open set $U \subseteq \mathbb{C}$. Let $V=\{z\in\mathbb C:\overline z\in U\}$. Define $g$ on $V$ by $g(z)=\overline{f(\overline{z})}$. Show that $g$ is analytic ...
3
votes
1answer
196 views

Prove that a function is subharmonic

I am trying to prove the following conjecture (or find a counterexample): Claim: Let $f(z)$ and $g(z)$ be holomorphic functions defined on a simply connected bounded domain $\Omega \subset \mathbb ...
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votes
0answers
57 views

u and v are real and imaginary. f(z) is a piecewise function. Show that Cauchy Riemann holds but is not differentiable at 0.

Say u and v are real and imaginary components. If given f(z) how would I show that Cauchy Riemann holds at z=0 but is not differentiable? Let f(z)=$(u-iv)^2$/z when z$\neq$0 and f(z)=0 when z=0 What ...
0
votes
1answer
55 views

Complex value of z for real and imaginary part and modulus

how do we go about if the function is $g(z) = e^{-2z}$ ? what values of $z$ makes $g(z)$ real, imaginary, or modulus <1? $g(z) = e^{-2(x+iy)} $ = $e^{-2x-i2y} $ = $e^{-2x}e^{i(-2y)} $ ...
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vote
1answer
37 views

Complex values of z for real and imaginary part

If we have $f(z) = e^{iz} $, what values of $z$ for which $f(z)$ is real, imaginary, or of modulus <1? We know that $e^{iz} = cos(z)+isin(z)$, hence, $f(z)$ is real if $sin(z)=0$, imaginary if ...
0
votes
2answers
30 views

Complex power series exponential

How do we evaluate $\sum _{n=0}$ $\frac{8^n}{(3n)!} $? We know that $e^z$= $\sum _{n=0}$ $\frac{z^n}{(n)!} $ So can we say that $\sum _{n=0}$ $\frac{8^n}{(3n)!} $ = $\frac{e^8}{3}$ ? Obviously the ...
1
vote
1answer
61 views

Power series of sum of complex exponentials

How do we express $e^z+e^{\alpha z}+e^{\alpha^2z}$ as a power series? Is it simply $$\sum \left[\frac{z^n}{n!} + \frac{(\alpha z)^n}{n!}+\frac{(\alpha^2 z)^n}{n!} \right]$$ or is it more complex ...
0
votes
1answer
23 views

How would you go about proving that any complex line is a curve on the Riemann sphere?

It is physically intuitive why a line (for example the complex line) in the complex plane would map to a circle (or the circumference) of the Riemann sphere, but how would you go about showing this ...
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0answers
31 views

Proof of exponential complex constant

If we let $ g(z) = e^{f(z)} $ and $f$ is holomorphic on a region G, how do we prove that $f$ is a constant if $g$ is a constant? Do we just simply find the derivative of $g(z)$ and set it to zero, ...
1
vote
1answer
95 views

Evaluating Complex Expressions

I need to evaluate the value(s) of the following expressions, giving my answers in the form $x+iy$. I believe I've attempted the first few correctly, but the last one I'm completely baffled on. a) ...
0
votes
1answer
35 views

Does there exist a notation for the set of poles of a function f(x)?

For eigenvalues we have a really nice notation $\sigma$ denoting the spectrum of this matrix i.e. the set of all eigenvalues. Before knowing $\sigma$, I just used $eigs(A)$ to denote the set of ...
1
vote
1answer
13 views

Injectivity/Complex Mapping

In $\mathbb{C}$, consider the neighborhood $O$ $:=$ {${x+iy; (x+1)^2 + (y-1)^2 < 4}$}. Show that $w=z^2$ is not one-to-one when its domain of definition is restricted to $O$. I'm not sure where to ...
0
votes
2answers
309 views

Complex Analysis: Show that $Arg(z)$ is discontinuous at each point on the nonpositive real axis.

That's the question. I'm not entirely how to prove, per se, that $Arg(z)$ is discontinuous at every point on the nonpositive real axis. By defintion, $- \pi < Arg(z) \leq \pi$, so I'm not sure ...
0
votes
2answers
81 views

Dirichlet's Kernel, Identity for complex numbers

It is easy to prove that $ 1/2 + \sum\limits_{k=1}^{n} \cos(kt) = \frac{\sin(n+1/2)t}{2\sin(t/2)} $ for every real number t. I know also that this identity doesn't hold for complex numbers but i ...
1
vote
1answer
337 views

Using the Residue theorem or Fourier transform to solve this integral

The following is an integral I am trying to evaluate $$I= \int_{-\infty}^\infty f(s) \, ds = \int_{-\infty}^\infty \frac{\frac{1}{(1- \ \ 2 \pi j a_1 s )^{3}(1-2\pi j a_2 s)^3}-1}{2\pi j s }\ ...
3
votes
1answer
28 views

Proving $||x||_{\infty}$ is a norm on $\mathbb{C}^n$

I'm trying to show that given $x,y \in \mathbb{C}^n$, the following holds $$ ||x + y||_{\infty} \leq ||x||_{\infty} + ||y||_{\infty} $$ Assume $x = a + ib, y = c + id, a,b,c,d \in \mathbb{R}$. $$ ...