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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22 views

Proving a property of a holomorphic function $ f $ on the unit disk that satisfies $ f(0) = 1 $. [on hold]

If $ f $ is a non-constant holomorphic function in the unit disc $ |z| < 1 $ that satisfies $ f(0) = 1 $, then prove that there are only finitely many points $ z $ lying inside the disc such that $ ...
2
votes
1answer
26 views

Computing Residues Confusion

For $C := \left \{ |z| = 2\right \}$, $\int_{C}\frac{e^{\pi z}}{4z^2 + 1}dz$ the isolated singularities are $\pm \frac{1}{2}i$. By Cauchy Residue Theorem,$\newcommand{\Res}{\operatorname{Res}}$ ...
0
votes
0answers
19 views

Analytic continuation at polygon vertices

The Riemann Mapping Theorem states that there exists a bijective, biholomorphic mapping from a simply connected set $\Omega \ne \mathbb{C}$ to the unit disk. Schwarz-Christoffel gives a (mostly) ...
-3
votes
2answers
70 views

Difficult Complex Number Proof. Given $|w| =1$ or $|v|=1$ [on hold]

Let $z, w$ be distinct complex numbers. Show that if $|z| = 1$ or $|w| = 1$, then $$\left|\frac{w-z}{1-\overline{w}z}\right| = 1$$ Hint: Note that $|a|^2 = a\overline a$ I have been ...
1
vote
1answer
51 views

Trigonometric integral evaluates to factorial

I would like to prove the integral identity $$\int_{0}^{2\pi} e^{\cos(x)} \cos(nx - \sin(x)) \, dx = \frac{2\pi}{n!}$$ One approach is to interpret this as the real part of a complex exponential ...
0
votes
1answer
33 views

Prove that $\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$

Let $a,b,c$ be complex numbers such that $|a+b|=m$ and $|a-b|=n$ and $mn\ne0$. Prove that $$\max\{|ac+b|,|a+bc|\}\ge\frac{mn}{\sqrt{m^2+n^2}}$$ I have tried using formula ...
0
votes
2answers
37 views

Cauchy- riemann equations

Let $f(z) = u(x,y) + iv(x,y)$ be a complex function that is differentiable at the point $z_0 =x_0 + iy_0$. Prove that $f'(z_0)= \frac{\partial u}{\partial x} (x_0,y_0) + i \frac{\partial ...
0
votes
0answers
25 views

Meromorphic Function [duplicate]

Let f be a meromorphic function on $\mathbb{C}$ such that $|f(z)| \geq|z|$ at each $z$ where f is holomorphic then f is entire finction such that $f(z)=Az$ for some constant $A \in \mathbb{C}$.
1
vote
2answers
44 views

Determining the Laurent Series

I need to determine the Laurent series of this function: $$\frac{1}{(z-1)(z+5)}$$ Inside the annulus: $$\left\{z|1<|z-2|<6\right\}$$ Any help appreciated.
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0answers
17 views

(Churchill Brown) Integration Branch Point Exercise [on hold]

I am having trouble with the following problem in Churchill/Brown text. I am not able to find a concrete answer. The problem is given below. The reference to Fig99 is completely unnecessary. I know ...
0
votes
1answer
37 views

If $f$ has a primitive on $\Omega$, then $f$ is analytic on $\Omega$

If $f$ has a primitive on $\Omega$, then $f$ is analytic on $\Omega$ I don't understand the proof of the corollary $2.2.12$ here. How can one apply corollary $2.2.11$, if it holds only for ...
1
vote
2answers
40 views

To find analytic function with given condition

How to find all analytic function on the disc $\{z:|z-1|<1 \}$ with $f(1)=1$ and $f(z)=f(z^2)$ ?.
0
votes
0answers
33 views

Expected Value of the absolute value of the sum of random variables

Hi everyone and thanks in advance. Let's say we have a random variable Y which can be expressed as the sum of two other complex random variables X and W, i.e. $ Y = X + W $. $X$ and $W$ are ...
1
vote
1answer
42 views

functions that can be written as $g^3$

Let $D$ be a proper sub-domain of $\mathbb{C}$ in which every everywhere nonzero function $f$ can be written as $g^3$ with $g$ being holomorphic, then show that there is a holomorphic embedding of $D$ ...
0
votes
1answer
24 views

Removable Singularities

Can we not we define every isolated singularity of a complex analytic function to be a removable singularity? A removable singularity is a point where the holomorphic function is undefined, but it ...
1
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0answers
30 views

How to find fourier transform of $e^{-x^2}$?

I want to find the fourier transform of $e^{-x^2} = \int_{-\infty}^{\infty}e^{ikx-x^2}\,dx$ using contour integration. I consider the rectangular contour $C$ with verticies $\pm R, \pm R + ik$ Then ...
0
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0answers
20 views

Applying Cauchy Residue Theorem

For $C := \{z(t) : t(1+i) : t \in [-1,1]\}$, $\int_{C} \frac{dz}{(z-1)}$. The singularities of $\frac{1}{(z-1)}$ is $z_0 = 1$. Note that this singularity (pole?) is contained within the contour. ...
1
vote
1answer
42 views

How to find $z$ with $|\sin z | \le 1$?

I am trying to find all $z \in \mathbb C$ such that $|\sin z |\le 1$. What I did so far: Clearly, for all real $z$ this is satisfied. Next I tried to rewrite it like this: $$ |\sin z |^2 = ...
1
vote
2answers
33 views

Simple Question About Residues/Poles/Zeros/Singularities

I'm having a little bit of trouble with residues. If we have the $f(z)=\left(\frac{\cos(z)-1}{z}\right)^2$ at $z_0=0$, we have a zero of order 2 in the numerator and a zero of order 2 in the ...
0
votes
0answers
84 views

Understanding Eigenvector

We have a matrix $A$ of size $N \times M$, where $N\le M$. Consider a vector $V$ of length $N$. Now I take product of $AV$ to get a vector $W$ of length $M$. Here I have projected the original ...
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votes
2answers
34 views

Compute the following integral, where $C$ is the circle $|z|=3$

Evaluate:$$\int_{C} (1 + z + z^2)(e^\frac{1}{z}+e^\frac{1}{z-1}+e^\frac{1}{z-2}) dz $$ where $ C$ is a circle $|z|=3$ and $z \ \epsilon \ \mathbb{C}$ The function that is being integrated has ...
0
votes
1answer
21 views

Show the Inner product equals the Hermitian Product

Let $\langle., .\rangle$ denote the usual inner product in $\mathbb R^{2}$. In other words, if $Z = (x_{1}, y_{1})$ and $W = (x_{2}, y_{2})$, then $\langle Z,W \rangle$ = $x_{1}x_{2} + y_{1}y_{2}$. ...
2
votes
1answer
20 views

Fourier Transform for Boundary Value Problems

I am trying to understand the problem defined by $$\phi_{xx} + \phi_{yy} = 0, in -\infty \lt x \lt \infty, y \gt 0$$ $$\phi = f(x) \space as \space y \to 0, \phi = 0 \space as \space y \to \infty$$ ...
0
votes
1answer
22 views

Brief Complex Analysis Problem

With $ω = se^{iϕ}$, where $s ≥ 0$ and $ϕ ∈ R$, solve the equation $z^{n} = ω$ in $C$ where $n$ is a natural number. How many solutions are there? What I have so far: $ln(z^n)=n ln(z)$=$ln(w)$ ...
3
votes
0answers
18 views

Using term-by-term Integration to solve LaPlace Transforms

I am attempting to use term by term integration to find the LaPlace transform of $$u(t) = \frac{sin(t)}{t}H(t)$$ The LaPlace transform is going to be $\int_0^\infty \frac{sin(t)e^{-st}}{t}$. Every ...
1
vote
1answer
47 views

Is a holomorphic function analytic in a ‘real’ sense?

I am taking a course in complex analysis, and I asked myself the following question: If a function $ f: \mathbb{C} \to \mathbb{C} $ is holomorphic, can its real and imaginary parts be given by a ...
1
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1answer
35 views

Complex Number - root

The complex numbers $z$ and $w$ satisfy $z^{13} = w$, $w^{11} = z$, and the imaginary part of $z$ is $\sin\left(\frac{m\pi}n\right)$ for relatively prime positive integers $m$ and $n$ with $m < ...
2
votes
5answers
83 views

Argument of $z = 1 - e^{it}$

Let $t\in(0,2\pi)$. How can I find the argument of $z = 1 - e^{it}= 1 - \cos(t) - i\sin(t)$?
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0answers
20 views

Velocity and Accelaration in the z and w planes

I am stuck on the following problem A particle $P$ moves along the line $x+y=2$ in the $z$-plane with a uniform speed of $3\sqrt 2$ feet per second from the point $z=-5+7i$ to $z=10-8i$. If ...
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votes
4answers
32 views

Square in the complex plane given three vertices. Find the fourth complex number vertice.

There is a square in the complex plane. Four complex numbers form the four vertices of this square. Three of the complex numbers are $-19 + 32i,$ $-5 + 12i,$ and $-22 + 15i$. Find the fourth complex ...
0
votes
1answer
20 views

Path Integral (Cauchy's Theorem)

An old exam style question I have encountered asks to evaluate the integral of the following; $\int_{C}\frac{1}{1+\sqrt{z+2}}$ where $C$ is the positively oriented unit circle. So firstly I ...
1
vote
1answer
23 views

Differentiating a purely imaginary function

If a function is defined as being the imaginary part of some expression, how do I take the derivative of the function? Do I: (a) Take the imaginary part of the expression, and then differentiate? ...
1
vote
1answer
32 views

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$

Show that there exists an entire function $h$ such that $\lim_{n\to\infty}{h(nz)}=0$ for all $z\ne0$. The following construction is in Walter Rudin's Real and Complex Analysis Chapter 16, Exercise 11. ...
0
votes
1answer
47 views

How can I solve this integral with complex number?

$n$ here is a complex number such that $n=n_r+in_i$ How can I solve this integral? $$\int_{0}^{\infty}\frac{x^4}{|x^2-n^2|^2} d x=? $$
3
votes
2answers
33 views

Computing $\int_{\gamma}e^zdz$, where $\gamma$ is a particular semicircle

How can I compute $\int_{\gamma}e^zdz$, if $\gamma$ is the semicircular arc depicted below? So, $\gamma=3e^{i\theta(t)}$, with $0\le\theta(t)\le\pi$, and then ...
1
vote
1answer
28 views

Contour integration for a ratio of a trig. function and a polynomial

We are supposed to use contour integration to determine $$\int_{-\infty}^\infty f(x)dx=\int_{-\infty}^\infty \frac{\cos(x)dx}{x^2+2x+5}$$ and we are given the (obvious) hint that ...
3
votes
1answer
63 views

Prove that $\Big|\frac{f(z)-f(w)}{f(z)-\overline{f(w)}}\Big|\le \Big|\frac{z-w}{z-\overline w}\Big|$

Let $\mathbb{H}$ denote the upper half plane of $\mathbb{C}$, i.e. \begin{equation*} \mathbb{H}=\{z \in \mathbb{C}: Im(z)> 0\} \end{equation*} Suppose $f:\mathbb{H}\to\mathbb{H}$ is analytic. ...
0
votes
1answer
18 views

Find the Laurent series for $f(z)=\frac{e^z}{(z-i)^4}$ at $z=i$.

Find the Laurent series for $f(z)=\frac{e^z}{(z-i)^4}$ at $z=i$. What I was thinking of using $e^z=\sum_{i=0}^\infty \frac{x^i}{i!}$. But from there I am not sure what to do?
2
votes
0answers
46 views

Series for $\sin(z) / \sin(\pi z)$

${\sin(z) \over \sin(\pi z)} = 1/\pi + {z \over \pi} \sum_{n \in {\bf Z} \setminus \{0\}} {(-1)^n \sin(n) \over n(z-n)}$ First I apply Mittag-Leffler's theorem, to see that the RHS is a mereomorphic ...
2
votes
2answers
42 views

An analytic function on the disk that sends the boundary into the boundary sends the interior onto the interior

Consider $f$ analytic on $B(0; 1) = \{z \in \mathbb{C}\ |\ |z| < 1\}$ and continuous in $D(0; 1) = \{z \in \mathbb{C}\ |\ |z| \leq 1\}$ such that $f(S^1) \subset S^1$. I must show that $f(B(0; 1)) ...
1
vote
1answer
33 views

Cauchy-Riemann equations in polar form

Show that in polar coordinates, the Cauchy-Riemann equations take the form $\frac{\partial u}{∂r} = \frac{1}r \frac{∂v}{∂θ}$ and $\frac{1}r \frac{\partial u}{∂θ} = −\frac{∂v}{∂r}$ . Use these ...
1
vote
2answers
75 views

$\int\limits_{\gamma} \frac{z}{(z-1)(z-2)}dz$, $\gamma(\theta) = re^{i\theta}$, $2 < r < \infty$

For $0 < r < 2$, we can use Cauchy's integral formula and choose our holomorphic function to be $f(z) = \frac{z}{z - 2}$ since $z = 1$ is the only pole, but if $r > 2$, then both poles $z = ...
0
votes
1answer
20 views

Transforming a function to use method of residues

Suppose I have an integral $$I=\iiint_{\mathbb{R}^3}\dfrac{d^3\textbf{k}}{(k^2+\gamma)^2}$$ where $\gamma$ is independent of k. $d^3\textbf{k}$ is given as the 3 components of a vector. I am asked ...
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vote
1answer
48 views

Stone Weierstrass and Runge

Suppose $E(closed)\subset\{z:|z|=1\}$ and let $f(z)$ be a continuous function on the set $E$. I want to show that $f(z)$ can be approximated by polynomials on $E$. I am not exactly sure how to solve ...
0
votes
1answer
60 views

$f$ is differentiable on $U\setminus\{p_1,\dots,p_r\}\implies$ $f$ is holomorphic on $U$

Let $U\subset\mathbb C$ be open and $p_1,\dots,p_r$ be finite number of points in $U$. If $f:U\to\mathbb C$ is continuous function that is complex-differentiable in any point of ...
1
vote
1answer
62 views

Basic complex integration question

If I have an integral: $$\int_{0}^{2\pi} \frac{1}{3+2cos(t)^2} = \frac{a\pi}{b}$$ How can I find a and b? What formula do I use?
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vote
1answer
32 views

Confusion with $\int_{C}\frac{ze^z}{z^6 - 1}$

I must solve $\int_{C}\frac{ze^z}{z^6 -1}dz$ where $c:= \{ z \; : \; |z-a|=a\}$ and $a > 1$. I wish to apply the Cauchy Integral Formula (or generalized). The only singularities inside the ...
0
votes
2answers
40 views

$\int_{\gamma}\left(\frac{1}{z}-\frac{1}{z-1}\right)dz=0$ on $\mathbb C\setminus[0,1]$

$\int_{\gamma}\left(\frac{1}{z}-\frac{1}{z-1}\right)dz=0$ on $U:=\mathbb C\setminus[0,1]$ for a closed path with image in $U$ For any analytic function $f$ and a closed path $\gamma$: ...
1
vote
0answers
50 views

Why can we write $\displaystyle u(x+h_1,y+h_2)-u(x,y)=\frac{\partial{u}}{\partial{x}}h_1+\frac{\partial{u}}{\partial{y}}h_2+|h|\psi_1(h)$?

In this image, why can we write $\displaystyle u(x+h_1,y+h_2)-u(x,y)=\frac{\partial{u}}{\partial{x}}h_1+\frac{\partial{u}}{\partial{y}}h_2+|h|\psi_1(h)$ ? [I borrowed link to the image uploaded by ...
2
votes
1answer
31 views

Show that $z^n+nz-1$ has $n$ zeros in $D(0,R)$

Let $n\geq 3$. Show that the polynomial $z^n+nz-1$ has $n$ zeros in $D(0,R)$, where $$R=1+\left(\frac{2}{n-1}\right)^{1/2}.$$ I was hoping to use Induction and Rouche's Theorem. For the base case ...