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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18 views

Prove that the point will go 3 times around ellipse

I'd like to prove that if a point $z$ goes once around ellipse with focus $2,-2$ then point $z^3-3z$ goes 3 times around some ellipse with the same focus. I was thinking (since ellipse is a set of ...
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2answers
36 views

Expanding a complex function in Taylor series

Expand the function $$ f(z) = \frac {2(z + 2)} {z^2 − 4z + 3} $$ in a Taylor series about the point $ z = 2 $ and find the circle C inside of which the series converges. Find a Laurent series that ...
3
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2answers
309 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
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1answer
164 views

If a holomorphic map from an open disc to $\mathbb{C}^n$ extends continuously to the closed disc, what about its partial derivatives?

Let $F$ be a holomorphic map from an open disc $D \subset \mathbb{C}^n$ to $\mathbb{C}^n$ and suppose $F$ extends continuously to $\overline{D}$. Do the maps $\partial F_i / \partial z_k$ extend ...
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1answer
13 views

Determining position of pole

i would like to know how to determine if pole of given function is inside a circle of radius 2? for example let us take this function $$ f(z)=1/\cos z $$ We have poles at $$ z=\pm \pi/2, \pm 3\pi/2, ...
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1answer
22 views

Find the radius of convergence for $\sum_{n=1}^{\infty}\frac{2^n}{3^n+4^n}z^n$

$$\sum_{n=1}^{\infty}\frac{2^n}{3^n+4^n}z^n$$ What I've done is try to evaluate the expression sans $z^n$ with the root test. $$\sqrt[n]{\frac{2^n}{3^n+4^n}}=\frac{2}{\sqrt[n]{3^n+4^n}}$$ But I'm ...
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1answer
3k views

Straight Line Equation in Complex Plane

I'm confused about the straight line equation in complex plane: how does $0 = Re((m+i)z + b)$ come from $y = mx + b$? I mean when I see $y = mx + b$, I can draw a graph in my mind, but when I see $0 ...
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1answer
63 views

How to express $\sin \sqrt{a-ib} \sin \sqrt{a+ib}$ without imaginary unit?

I got this kind of expression as a value of an infinite product: $$\prod_{k=1}^{\infty} \left(1-\frac{A}{k^2}+\frac{B}{k^4} \right)$$ It's easy to see how it can be factored into a product of two ...
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22 views

Difficulty in Laurent series

I have to find the order of pole of $$f(z)=\frac{\sinh z}{z^7}$$ after expansion of this function I get $$f(z)=1/z^6+1/3!z^4+1/5!z^2+1/7!+z^2/9!$$ It contains only three term in the ...
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3answers
60 views

$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$, $\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$ - Cauchy integral?

Compute the integrals $$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$$ and $$\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$$ I am stucked on these problems since a good while. Is there someone is able ...
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1answer
21 views

Show the following series converges uniformly using Weierstrass M Test

I'm trying to show that the following series converge uniformly by using the Weierstrass $M$ Test. $$ \sum ^{\infty}_{j=0}z^{n},\ \ \ 0\leq \left | z \right |< R,\ \ \ R<1 $$ and $$ \sum ...
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1answer
34 views

Does tanz assumes all complex numbers? [on hold]

$f(z)=tanz$,$z$ $\in$ $C$ a)assumes all complex numbers b)assumes none of complex numbers c)assumes all complex numbers except $i$ d)assumes all complex numbers except $i$ and $-i$ I think ...
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3answers
540 views

The solution of Cauchy-Riemann equation

Can you give me an example that there is a $f \in C_0^{\infty}(\mathbb C)$, such that the equation $\bar \partial u=f$ has no $C_0^{\infty}(\mathbb C)$ solution?
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24 views

Find the isolated singularities of $\displaystyle f(z)=\frac{1}{3+\sqrt{1+\sqrt{z}}}$ and classify them.

Find the isolated singularities of $\displaystyle f(z)=\frac{1}{3+\sqrt{1+\sqrt{z}}}$ and classify them. I am not sure how to find isolated singularities when I have a square root function. Any ...
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0answers
29 views

Proving the function is holomorphic and jumping

This is the problem 5 in chaper 3 in Stein-Shakarchi's complex analysis. It states let $g(z)=\frac{1}{2\pi i}\int_{-M}^M{\frac{h(x)}{x-z}}{dx}$ where $h$ is continuous and supported in $[-M,M]$ Now, ...
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1answer
14 views

What is the Laurent expansion of f(z)=1/(z-3)?

What is the Laurent expansion of f(z)=1/(z-3)? In the region, ㅣZ-3ㅣ>0 ? I just computed the Laurent expansion in the region ㅣZㅣ>3 by dividing the denominator by 1/z and making it as a geometric ...
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1answer
19 views

Expand the Laurent series

Expand $f(z)= \frac {z}{(z+1)(z-2)}$ in a Laurent series valid for the given annular domain: $0 \lt \lvert z+1 \rvert \lt 3$ I'm having a lot of trouble with this one. The answer, per the back ...
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16 views

limits and convergence of sequences complex

For the following sequence discuss its limits and whether the convergence is uniform, in the region $\alpha \leq \left | z \right |\leq \beta $, for finite $\alpha$,$\beta >0$. $$\left \{ ...
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2answers
51 views

Finding the sum of a complex series

Find the sum of the series: $$ \sum_ {n=1} ^{\infty} {nz^n} , $$ $$ |z| < 1$$ Where do I start from? Can I use the root test?
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1answer
518 views

Proof of Laurent series co-efficients in Complex Residue

Am trying to see if there is any proof available for coefficients in Laurent series with regards to Residue in Complex Integration. The laurent series for a complex function is given by $$ f(z) = ...
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1answer
44 views

Riemann Zeta Function for $\Re(s)=0$

All the sources I have read talk about continuation from $Re(s)>1$ to $Re(s)>0$ then $Re(s)<0$ $(s\neq 1)$. What about $Re(s)=0$? Where does that go?
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34 views

Using the $\cot (\pi z)$ to find $\sum \frac{1}{n^2}$ [duplicate]

I'm trying to prove the result that $$\sum_1^\infty \frac{1}{n^2}=\pi^2/6$$ using cotangents and residue theory. I know that $\sum f(n)=-$Sum of residues of $\pi \cot (\pi z)f(z)$ at the poles of ...
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16 views

Understanding exponential function [on hold]

It´s me again. Consider $\phi:\left(-1,1\right)\longrightarrow\Bbb S$ \ $\lbrace-1\rbrace$ where $\Bbb S=\lbrace z\in \Bbb C: \vert\vert z\vert\vert=1\rbrace$ $\phi\left(t\right)=e^{i\pi t}$ My ...
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1answer
434 views

Prove the complex function is entire

The function $$ f(z) = e^{x^2 - y^2} (\cos 2xy + i \sin 2xy )=e^{z^2} $$ Then, how to prove it's analytic everywhere of complex plane of the exp function...?
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1answer
24 views

How can I show this homotopy is continuous?

this is the homotopy that transforms the unit circle into the unit square in $\mathbb{C}$. The function is defined by $h(t,s) = (1-s)e^{2\pi it} + s$*$\{$... a piece-wise function consisting of ...
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2answers
44 views

Singularity of $f(z)=\frac{\sin z}{z}$ at $z=0$

I'm reading Conway's complex analysis book and on page 110 he asked to determine the nature of the singularity at $z=0$ of the function $f(z)=\frac{\sin z}{z}$ and if it's a removable singularity he ...
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7 views

Deformation of Gamma function integral contour

Terence Tao has described the gamma function as the inner product of a multiplicative and an additive character with respect to the Haar measure on $\Bbb R^+$. The gamma function is defined as ...
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2answers
40 views

A human way to simplify $ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 \sqrt{a^2 - 1}} - 2 a $

I end up with simplifying the following fraction when I tried to calculate an integral(*) with the residue theory in complex analysis: $$ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 ...
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3answers
43 views

Complex Analysis - what makes a simple connected set?

Having difficulty finding the differences between a connected set and a simply connected set and a region. Would be good if someone could inform me and also give an example. Thanks Tom
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1answer
24 views

Determine the Laurent expansion of $f(z)=\frac{z}{(z-1)(2-z)}$ for different regions in the complex plane.

I have tot determine the Laurent series of $f(z)=\frac{z}{(z-1)(2-z)}$ for the regions $|z-1|>1$ and $0<|z-2|<1$. I already know what to do for the regions $|z|<1$, $1<|z|<2$ and ...
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1answer
75 views

How to construct a polynomial with minimum deviation from zero on the complex region?

I need to compute the analog of Chebyshev polynomials (which give the minimum deviation from zero on [-1,1]) on the given region $\Omega\subset \mathbb C$. More precisely: find $P_n$ such that ...
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0answers
38 views

Inequality with complex numbers involving 6-th and 7-th root [on hold]

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$
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9 views

How would one show that a function $h(t,s)$ defines a homotopy?

Let's say I'm given a function $h(t,s)$ that deforms a closed path $c_1$ into another closed path $c_2$. Would it be enough to check that $h$ satisfies the three conditions: $h(t, 0) = c_1$, $h(t, ...
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1answer
37 views

Singular points of $\displaystyle \sin \left( \frac{1}{\cos\frac{1}{z}}\right)$

Specifically, $\displaystyle f(z) = \sin \left( \frac{1}{\displaystyle \cos \frac{1}{z}} \right)$ has singular points at $z = \displaystyle \frac{2}{\pi + 2\pi k}$, among others. Now, I am trying to ...
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1answer
62 views

Factoring semiprimes cost estimation

I have two problems that are the following. The first problem is the following: I need to estimate the cost of factorizing a given semiprime based on previous estimations. For example I have the time ...
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9 views

Arbitrary semi-circular path in the complex plane.

I want to try and define a path that starts at $\alpha$ and ends at $\beta$, but gets there by travelling on a circle, anticlockwise. so $\gamma(t) = \frac{\alpha + \beta}{2} + \frac{|\alpha - ...
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1answer
16 views

Discrepancy between text's answer and mine: singular points of $\cot\left(\frac{1}{z}\right) - \frac{1}{z}$

The points $\frac{1}{k\pi}$, where $k \in \mathbb{Z}$ are all singularities of the function $f(z) = \cot\left(\frac{1}{z} \right) - \frac{1}{z}$. My textbook seems to think that they are simple ...
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0answers
31 views

Let $P_1,P_2, \ldots,P_n$ be $n$ arbitrary points of the plane.

Let $P_1,P_2, \ldots,P_n$ be $n$ arbitrary points of the plane. If a variable point $P$ is confined to a closed bounded set $E$, show that the product $$\prod_{k=1}^n \overline{PP_k}$$ attains its ...
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3answers
396 views

Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot ...
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0answers
26 views

Find the derivative of complex function $z^{2i}$ at $z=i$ [on hold]

Let $z^\alpha$ represent the principle value of the complex power defined on the domain |z|>0, $-M<\arg(z)<M$ , Find the derivative of complex function $z^{2i}$ at $z=i$.
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1answer
79 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in ...
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1answer
30 views

What is the best way to solve $\lim_{n\to \infty}{(e^{i \theta})^n}$?

What is the best way to solve the limit: $\lim_{n\to \infty}{(e^{i \theta})^n}$ $\theta$ is fixed, but you must have a care for cases $\ \theta > 0 , \ \theta = 0 , \ \theta < 0.$ There ...
3
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1answer
304 views

Using Morera's theorem to prove analyticity

I have a function $F(x,y)$ which is continuous and analytic on the complement of a certain function $x(y)$. Is it possible to use Morera's theorem to show that it is analytic everywhere? Clearly, this ...
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1answer
111 views

Dual of holomorphic functions (with the $L^1$ topology)

Let $\Omega$ be a connected domain of the complex plane, and let $E$ be the vector space of integrable holomorphic functions on $\Omega$. Then it can be checked that $E$ is a closed subspace of ...
3
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1answer
55 views

Singular point of $f(z)$ also a singular point of $1/f(z)$ and $f^{2}(z)$

Suppose $z_{0} \in \mathbb{C}$ is an isolated singular point of the function $f$ of a given type (removable, pole of order $N$, essential). I need to show that $z_{0}$ is an isolated singular point of ...
3
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1answer
122 views

Prove that $f(z)=z$ on region $D$ [duplicate]

Given that $D$ be a bounded region containing $0$ and $f:D\rightarrow D$ be a holomorphic mapping such that $f(0)=0,f'(0)=1$ prove that f(z)=z for all $z$ in $D$ This problem reminded me of the ...
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1answer
24 views

Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually ...
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0answers
17 views

How do I find an upper bound on $z/(z^3+1)$ on a circular path with radius R centred at the origin?

I want to use the estimation theorem, so I want to find an $M$ such that $|\frac{z}{z^3 + 1}| < M$ I cant seem to work with the $z^3$. $$|z^3 + 1| \geq |z^3 - 1| $$ is just not true. How can I ...
0
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0answers
19 views

principle laurent series of $f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)}$ at $z=0$ and determine $\oint_{|z|=1} f(z)dz$.

Question: So given the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)}.$$ First: Give the principal part of the Laurent series of $f$ at $z = 0$. Second: Determine the integral $\oint_{|z|=1} ...
0
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0answers
25 views

Show that there can only be finitely many numbers $n \in \mathbb{N}$ for which $f(1/n) = 1/(n + 1)$. [duplicate]

Let $f : \mathbb{C} → \mathbb{C}$ be analytic. Show that there can only be finitely many numbers $n \in \mathbb{N}$ for which $$f(1/n) = 1/(n + 1).$$ For this problem I was thinking about using the ...