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, ...

learn more… | top users | synonyms (2)

1
vote
2answers
64 views

Finding the Laurent Series for $\frac{1}{e^z-1}$ for $0<|z|<2\pi$

Since $\left|\dfrac{1}{e^z}\right|<1$ I figured I could rewrite the given function into a geometric series: $$\sum_{n=1}^{\infty} \frac{1}{(e^z)^n}$$ But this seems to be way off the mark. I ...
0
votes
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, ...
1
vote
1answer
26 views

Find the first terms of the Laurent series for: $\frac{e^{\frac{1}{z}}}{z^2-1}$

$\frac{e^{\frac{1}{z}}}{z^2-1}$ for $|z|>1$ I factored out the denominator and rewrote it to a geometric series and got the following expression: ...
0
votes
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 ...
1
vote
2answers
33 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 ...
-1
votes
1answer
46 views

Inequality of complex numbers involving modules [duplicate]

Let $z \in \Bbb C$ such that $|z| \ge 1$. Show that $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$ My try: I wrote $|z|^2$ as $z\times \bar z$, but I didn't get to any result. Can ...
4
votes
0answers
74 views
+50

Proof of an inequality in $\mathbb{C}$

Let $z\in \mathbb{C}, n \geq 2$. Show this complex inequality $$|z^n-1|^2\le |z-1|^2\left(1+|z|^2+\dfrac{2}{n-1}\Re{(z)}\right)^{n-1}$$ For $n=2$ the inequality is easy to prove: $$|z^2-1|^2\le ...
2
votes
3answers
50 views

How do I prove that $\lim_{z\to i} z^2=-1$?

How do I prove the following limit using the limit definition? $$\lim_{z\to i} z^2=-1$$ Using the limit definition $$|z^2+1|<\epsilon, \;\text{whenever} 0<|z-i|<\delta$$ so I factor out to ...
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
0answers
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 ...
2
votes
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, ...
1
vote
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 ...
0
votes
1answer
33 views

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

Find and classify all isolated singularities of $\displaystyle f(z)=\frac{1}{1+\sqrt{z}}$. So if $1+\sqrt{z}=0$ then $\sqrt{z}=-1$. Therefore $z=1$. Hence, $1$ is an isolated singularity and it ...
1
vote
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 ...
0
votes
0answers
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 \{ ...
2
votes
1answer
47 views

Show that $\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}…\cot \frac{(m-1)\pi}{2m}=1$

Prove: $$\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}...\cot \frac{(m-1)\pi}{2m}=1$$ This is a roots of unity problem. I managed to show a similar example for $\cos$ by the ...
1
vote
2answers
49 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?
4
votes
2answers
98 views

Evaluation of $ \int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx $

The following is an exercisein complex analysis: Use contour integrals with $-\pi/2<\operatorname{arg} z<3\pi/2$ to compute $$ I:=\int_0^\infty\frac{x^{1/3}\log x}{x^2+1}\ dx. $$ I don't ...
0
votes
1answer
42 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?
-1
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
1answer
23 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 ...
0
votes
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 ...
0
votes
0answers
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 ...
0
votes
3answers
42 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
1
vote
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 ...
0
votes
0answers
57 views

Finding a conformal map to the upper half-plane

Find a conformal map from the set $$\{z \in \mathbb{C}: |\operatorname{Im}z| < \pi \}\setminus \left[-\pi i; 0 \right]$$ to the upper half-plane. I have used a composition of the following maps: ...
0
votes
1answer
23 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 ...
0
votes
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 $$\sqrt[6] \frac {|2z-1|^2} {7} \ge \sqrt[7] \frac {|z-1|^2} {3}.$$
0
votes
0answers
8 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, ...
1
vote
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 ...
0
votes
0answers
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 - ...
0
votes
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 ...
0
votes
0answers
16 views

Contour integral along an ellipse

The question reads: Evaluate $$ \int_\gamma f(z)dz, f(z)=2x-3iy: x,y \in \mathbb{R} $$ where $$\gamma(t)=cost+2isint: 0\leq t\leq 2\pi $$ I know that the path formed is an ellipse, but this is the ...
1
vote
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 ...
0
votes
0answers
24 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 $z=i$.
1
vote
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 ...
9
votes
3answers
395 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 ...
0
votes
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 ...
-1
votes
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 ...
0
votes
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
votes
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 ...
3
votes
1answer
54 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 ...
0
votes
0answers
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 ...
0
votes
1answer
13 views

Finding the modulus of complex functions

Let $\gamma$ be the path$$\gamma:\left[0,1\right]\rightarrow\mathbb{C}, t\rightarrow\exp\left(t+it\right)$$ I have found that $$\gamma'\left(t\right)=\left(1+i\right)\exp\left(t+it\right)$$ To find ...
0
votes
2answers
39 views

$\int^{2 \pi}_0 \frac{1}{3+2 \cos t}dt$ using $\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$ or using $u=\tan \frac{t}{2}$

Question : Compute the integral of $$\int^{2 \pi}_0 \frac{1}{3+2\cos t}dt$$ I am stucked on this problem since a good while. I think we could convert that real integral into complex integral and ...
1
vote
0answers
33 views

Prove the inequality about $Re(z)$

Consider three different vectors $x$,$y$ and $z$ in $\mathbb{C}^{n}$. So $x = (x_{1} \dots x_{n})$ and this is the same for $y$ and $z$. Now we have $\langle x,y\rangle = ...
0
votes
1answer
46 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
0
votes
0answers
27 views

Necessity of Differential Forms

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis). Among them, some books also introduce ...