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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41 views

Want to prove certain sum representation of $\cot(x)$

So here is my problem, I would like to prove an identity I found in a book which was given without a proof. Namely $$-i\sum_{n\in\mathbb Z} \operatorname{sign}(n)\cdot e^{i2\pi nx}=\cot(\pi x)$$ I ...
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1answer
30 views

Showing that a function is holomorphic

Let $X \subset \mathbb{C}$ be a domain, $f : X \rightarrow \mathbb{C}$ continuos and for all closed triangles $\Delta \subset X$ applies $\int_{\partial\Delta}f \; dz = 0$. Show that $f$ is ...
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1answer
49 views

Using Cauchy's integral formula to calculate an integral

Calculate $\int_0^{2\pi} e^{\cos(x)} \cos(\sin(x)) \; dx$ using Cauchy's integral formula. I am really confused as I cannot bring the integral of the exercise and Cauchy's integral formula ...
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9 views

Show that $F_{ll}^{*}=\text{Re}g'_0-2l\text{Re}f_1+F_{ll}+\ldots$

I have stuck when I try to show (4.4): With $j \ge 1$, we have (4.4): \begin{align*}F_{ll}^{*}&=\text{Re}g'_0-2l\text{Re}f_1+F_{ll}+\ldots\\ ...
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1answer
56 views

Help with proving that this series diverges

I have this complex series $\sum\limits_{n=1}^{\infty} \frac{i^n}{\sqrt{n}}$, which I'm trying to prove converges. Now, I know that a complex sequence converges iff both its real and its imaginary ...
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1answer
53 views

Problem on solving Cauchy's Integral Formula [closed]

Question :- $$\oint_C\frac{1}{(z-z_0')(z-z_1')}dz = 0$$ for every simply closed contour $C$ that encloses the points $z_0'$ and $z_1'$. now My question is what is the value of $z_0'$ and $z_1'$ ...
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1answer
48 views

using Maclaurin series to estimate $\frac{1}{e} $

The problem I'm trying to solve is: Determine how many terms of the Maclaurin series of $f(x) =e^{-x}$ should be used to estimate $\frac{1}{e} $ with an error of magnitude less than $5 \times ...
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1answer
27 views

Complex Logarithm Derivation

I don't understand how the definition of the complex logarithm was derived. It is $ log(z) = ln|z| + i Arg (z) $, where $ z = x + iy $. I've tried all sorts of method to find this definition but ...
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25 views

Validity of $\int_{a}^b f(z) dz = \int_{a}^{\infty} f(z) dz - \int_{b}^{\infty} f(z) dz$?

What should be the conditions on the complex-valued function $f$ to be able to write : $$\int_{a}^b f(z) dz = \int_{a}^{\infty} f(z) dz - \int_{b}^{\infty} f(z) dz $$
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60 views

Prove that $e^{\ln{z}}=z$ from the power series

For my course in complex analysis we have to prove that the trivial relation $e^{\ln{z}}=z$. We are given the series for $\ln z$: $$f(w)=\sum_{n=0}^\infty (-1)^{n+1}\frac{w^n}{n}$$ $$\ln z = ...
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0answers
31 views

Find the domain of definition of a complex power function.

I am to find the domain of definition of a complex power function$$(z^2-4)^i$$ where it's branch cut is $L\frac\pi 2 $, that is $$ Arg(z) \in \left[\frac\pi 2,\frac5 2\pi\right] $$ I made it do ...
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2answers
208 views

Partitions of Natural Numbers [duplicate]

This is a question from Complex analysis by Stein. The question is Prove that it is not possible to partition $\mathbb N$ into finitely many infinite AP's with distinct common differences.(other ...
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1answer
57 views

Complex number condition on the modulus

The set of all complex numbers $(z_1,z_2)$ which satisfy $$\frac{|z_1 -z_2|}{|1-\overline{z_1}z_2|} \lt 1 $$ is? (Here $\overline{z_1}$ is $z_1$'s cojugate.) I attempted to write $z_1$ a as $x_1 + ...
4
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1answer
203 views

Binomial Summation

The sum $$ 1 + {n \choose 1}\cos \theta + {n \choose 2}\cos 2\theta + \cdots+ {n \choose n}\cos n\theta $$ is? I try to write this as the real part of $(1 + \cos \theta + i\sin \theta)^n$ but then ...
4
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1answer
105 views

Recommend textbooks that expain branch cut, Riemann surface and contour integration with branch cut in detail

I read several textbook on complex analysis, but few of them explain the branch cut and Riemann surface in detail and treat the contour integration with branch cut. But this is very important for many ...
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4answers
140 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
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1answer
73 views

Convergence of $\frac{1}{2^k} \frac{1}{z-w_k}$

Suppose $w_1,w_2,w_3,...$ are points on the unit circle. Consider the infinite series $$\sum_{k=1}^{\infty} \frac{1}{2^k} \frac{1}{z-w_k}$$ Let $D=\{z \in \mathbb{C}: |z|<1 \}$ A) Show that series ...
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1answer
64 views

Showing that the curves defined in the $xy$ plane by $u(x,y)=1$ and $v(x,y)=1$ cross at right angles at the origin.

Suppose $f$ is an entire function with $f(0)=1+i$. Let $u(x,y)=Re(f(x+iy))$ and $v(x,y)=Im(f(x+iy))$. A) Show that the function $u$ is a harmonic function of $x$ and $y$. B) Show that the curves ...
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47 views

Properties of power series and their analytic continuation

Suppose a power series $$\sum_{k=0}^\infty a_k z^k$$ is valid for $|z|<R$, and can be analytically continued to some function $f(z)$, for all $z\in\mathbb{C}$ , except for a finite number of points ...
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2answers
41 views

If $8iz^3 +12z^2 -18z +27i =0$ find the value of $4|z|^2$

Problem : If $8iz^3 +12z^2 -18z +27i =0$ find the value of $4|z|^2$ where $z$ is a complex number. Working : Now let $z = x+iy$ then $8i (x+iy)^3 +12(x+iy)^2 -18(x+iy) +27i =0$ $\Rightarrow ...
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1answer
33 views

How to find the locus of the complex equation : $z \overline{z} +az +b\overline{z}+c=0$

How to find the locus of the complex equation : $z \overline{z} +az +b\overline{z}+c=0$ I have no clue how to find the locus of such equation in complex plane. Please guide on this thanks in ...
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2answers
15 views

$|z_{1}- z_{2}| \leq |w_{1}- w_{2}| \implies |c_{1}z_{1}- c_{2}z_{2}| \leq |c_{1}w_{1}- c_{2} w_{2}|$?

Let $z_{1}, z_{2}, w_{1}, w_{2} \in \mathbb C$ with $|z_{1}- z_{2}| \leq |w_{1}- w_{2}|.$ Fix $c_{1}, c_{2}\in (0, \infty).$ My Question is: Can we expect, $|c_{1}z_{1}- c_{2}z_{2}| \leq ...
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3answers
171 views

Why is this polynomial a monomial?

Let $p$ be a polynomial of degree $n$ such that $|p(z)| = 1$ for all $|z| = 1$. Why is it that $p(z) = az^n$ for some $|a| = 1$? I've noticed that we could easily prove this by induction if we ...
2
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1answer
43 views

Fixed point of the map $z\mapsto e^{z-r}$.

I need to show that if $r>1$, then the map $z\mapsto e^{z-r}$ has exactly one fixed point in the half plane $\Re(z) <1$. I have tried setting $x+iy = e^{x-r}e^{iy}$ then compare real and ...
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1answer
28 views

Bound Function / Exponential (Unit Circle)

I need to mathematically prove the next inequality: $$\frac{(e^{2x}-1)^2}{(e^{2x}+1)^2} \le 1$$ If I graph the function, it is bounded above by 1, I don't know however how to proceed with this ...
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1answer
22 views

How can I prove the result $\int^a_{-a} f(x) dx = \int^a_0 f(x) + f(-x) dx$?

Suppose $f: [-a, a] \subset \mathbb R \rightarrow \mathbb C$. How can I prove the result $$\int^a_{-a} f(x) dx = \int^a_0 f(x) + f(-x) dx$$ ? I'm aware, that if I prove the result in the real case, ...
1
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1answer
67 views

Analytic maps from $\mathbb D$ to $\mathbb D-\{1/2,1/3\}$

Let $F$ be the family of analytic maps from the unit disk $\mathbb D$ to $\mathbb D-\{1/2,1/3\}$, then can we find a constant $M<1$ such that for any $f\in F$, $|f'(0)|\leq M$?
5
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2answers
135 views

Integral $\int_0^1 \frac{x\log x+1-x}{x \log^2 x}\log(1+x)\, dx=\log\frac{4}{\pi}$

Hi I am trying to prove this $$ I:=\int_{0}^{1} {x\log\left(\,x\,\right) + 1 - x \over x\log^{2}\left(\,x\,\right)}\, \log\left(\,1 + x\,\right)\,{\rm d}x=\log\left(\,4 \over \pi\,\right). $$ Thanks. ...
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1answer
19 views

Image of Circular Arc under composition of complex mapping.Help

Find the image of the circular arc $|z|=2,\ \ 0 \leq Arg(z) \leq \frac{\pi}{2}$ under the composition $f(z) = (h\circ g)(z)$ where $h(z) = \frac14e^{i\pi/4}z$ and $ g(z) = z^2$ This is what I have ...
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128 views

A uniformly convergent series

How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? ...
4
votes
1answer
48 views

Rouché theorem in queuing theory

I was looking for the uses of Rouché's theorem, and I came across queuing theory. An article stated that it is a workhorse theorem in this field, but as much as I tried to find some examples on the ...
2
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2answers
73 views

Riemann surface associated with complete analytic function of $(z^2-1)^{1/3}$

I'm trying to define an analytic function on '$\mathbb{C}$' of the form $f(z)=(z^2-1)^{1/3}$, i.e. I first remove two semi-infinite rays $l_1$ and $l_2$, one going from $1$ to $\infty$ along the ...
4
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1answer
43 views

Power series with differentiable coefficients

Suppose for each $s$ in an open interval, $P_s(x)=\sum_{k=0}^\infty a_k(s) x^k$ is a power series with radius of convergence greater than R, where each $a_k(s)$ is differentiable. My question is: Is ...
2
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3answers
45 views

Finding the image of a disk

Find the image of the disk $D= \{ z \in \mathbb{C}: |z|<1 \}$ under the mapping $z \to w=\frac{1}{z-1}$. My attempt: First I graphed the boundary. When $z=i, f(z)=-i/2-1/2$ When $z=-i, ...
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36 views
1
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1answer
68 views

Prove that if $f=u+iv$ is entire and $u(z) > v(z)$ for all $z$, than $f$ is constant.

Prove that if $f=u+iv$ is entire and $u(z) > v(z)$ for all $z$, than $f$ is constant. Could anyone give a rough proof of this? I am studying for a test and this problem is a practice problem and ...
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28 views

Can anyone prove this identity without passing through the complexified tangent space?

Let $\rho: \mathbb{C} \to \mathbb{R}$ be a smooth function, $\Omega = \{ z : \rho(z) <0 \}$, and suppose $|\nabla \rho| = 1$ on $b\Omega$. It is true that $$\int_{b\Omega} f(z) d\bar{z} = -2i ...
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1answer
24 views

Component of an Open Set: Polygonal Arcs

So I'm reading my Complex Analysis book, and I'm a little confused by the reasoning in this passage. I'll ask my question below first and then give the passage. Specifically, I'm puzzled by why it is ...
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1answer
55 views

Finding an upper bound for $|\int_{\gamma}e^{1/z}dz|$.

Find the upper bound for $|\int_{\gamma}e^{1/z}dz|$ where $\gamma$ is the part of the circle $|z| = \sqrt{8}$ from $2+2i$ to $-\sqrt{8}$. This is a question my professor went through quickly during ...
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1answer
47 views

An extension of Schwarz Lemma in Complex Analysis

Suppose that $f: D \to D$ is a holomorphic function on the open unit disc and $f(0)=0$. If we set $\displaystyle \omega=\exp\left(\frac{2\pi i}{n}\right)$ for $n \in \mathbb{N}$ then ...
2
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1answer
36 views

Finding Complex Zeros

I have to find how many zeros $3e^z - z$ has in $abs(z) < 1$. I know a function has a zero of order m if $f(z) = (z-z_0)^mg(z)$, where $g(z)$ does not equal 0. I was thinking of maybe applying ...
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1answer
44 views

Need some help finding this function.

I need to find a holomorphic function$f:\mathbb{C}\setminus\{0\}\mapsto \mathbb{C}$ with $\Re{(f)}=\dfrac{x+y}{x^2+y^2}$ and $f(1)=1$. I have gotten very stuck on this. Here is my working so far: ...
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1answer
77 views

Prove that $\arcsin z = \frac{\pi}{2} - \arccos z$

I have $\arccos (z) = -i\ln (z + \sqrt{z^2-1})$ and $\arcsin (z)=-i \ln(iz +\sqrt{1-z^2}).$ Now I must prove, that $\arcsin (z) = \frac{\pi}{2} - \arccos (z)$. I get: $$\arcsin ...
1
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1answer
24 views

Real-valued Discrete Fourier Transform

I have sequence of $N$ real numbers: $\mathbf{x} = (x_0, x_1, \ldots, x_{N-1})$. Discrete Fourier Transform (DFT) is defined as $$ X_k = \sum_{n=0}^{N-1} x_n e^{-i 2\pi k \frac{n}{N}}, \quad ...
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1answer
16 views

Let $f(x) := |\cos \frac x 2 |$ for $x \in \mathbb R$. Show by using Eulers formula that the $n$'th Fourier coefficient $c_n$.

Let $f(x) := |\cos \frac x 2 |$ for $x \in \mathbb R$. Show by using Eulers formula that the $n$'th Fourier coefficient $$c_n = \frac 1 {2\pi} \frac {(-1)^{n-1}} {n^2-\frac 1 4}$$ with respect to the ...
1
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1answer
170 views

About normal family of analytic functions

Suppose $G$ is a connected domain in $\mathbb{C}$ and $0,1\in G$.Let $\mathbb{F}$ be the family of analytic functions $f$ defined on $G$,$f(0)=0$ and $|f|<2$ on G.Prove that there is $0<c<2$ ...
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0answers
74 views

Paley Wiener theorem for entire functions of type $0$

We suppose that we have an entire function $E(k^2)$ of $k^2$ of order $1/2$, $(E(k^2)\leq Be^{C|k^2|^{1/2}})$. Then is it true that $E$ is exponential type $0$? If this is true then provided that ...
1
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0answers
22 views

Cauchy's formula with singularities on the curve

One knows that the Cauchy's formula asserts that for a given open set $\Omega\subseteq\mathbb{C}$ and $\gamma$ is a simply connected curve contained in $\Omega$, then ...
1
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0answers
20 views

Let $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show uniform convergence against $f$.

Let $\sum^{\infty}_{-\infty} |d_n| < \infty$ and define $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show it converge uniformly on ...
1
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2answers
26 views

Entire Complex Functions and Cauchy Integral Formula

Suppose that f(z) is an entire function. Show that $\exists c\in \mathbb{R}, c>0$ such that $|f(z)|\leq c|z| \forall z\in \mathbb{C} \Longrightarrow f(z)=az, a\in \mathbb{C}$ and $|a|\leq c$. ...