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
1answer
263 views

finding an image of a linear transformation

I am so confused in how its asking of finding an image of infinity. I am in my complex class and we have a test and this was part of the past midterm, I feel like if I do one all of the other ones ...
2
votes
1answer
178 views

Inversion in complex variables.

Hey guys I am given this question on my past midterm, and I cant come about the solution, i know its a mapping of all complex numbers minus the 0 and it maps to itself. So i tried to graph the points ...
3
votes
2answers
254 views

Find a branch for $(4+z^2)^{1/2}$ such that it is analytic in the complex plane slit along the imaginary axis from $-2i$ to $2i$

Find a branch for the multiple-valued function $(4+z^2)^{1/2}$ such that it is analytic in the complex plane slit along the imaginary axis from $-2i$ to $2i$ Also, isn't this function already ...
2
votes
1answer
146 views

Can I use Schwartz's Lemma to prove that $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ implies $f(z)=0$ for all $z\in\mathbb{C}$?

Problem. Suppose that $f(x)$ is an entire function satisfying $f(0)=0$ and $\operatorname{Re}f(z)\rightarrow 0$ as $|z|\rightarrow \infty$. Show that $f(z)=0$ for all $z\in \mathbb{C}$. The ...
6
votes
2answers
564 views

All the zeroes of $p(z)$ lie inside the unit disk

Let $p(z) = c_0 + c_1z + c_2z^2 + \dots + c_nz^n$ where $0 \le c_0 \le c_1 \le \dots \le c_n$. I would like to show that all zeroes of this polynomial lie inside the unit disk by applying Rouche's ...
2
votes
1answer
719 views

Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.

I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
2
votes
1answer
74 views

Open map in complex numbers

Is a function $f: \mathbb{C} \rightarrow \mathbb{C}$ where $f(z) = z^2$ an open map if $\mathbb{C}$ has the metric topology $d(z,a) = |z-a|$ ? I can think of several reasons why $f$ should map open ...
0
votes
1answer
48 views

Continuous complex funtion

I have this function $$F(z)=\frac{1}{\alpha-i\sqrt{z}}$$ with $\alpha>0$ and the determination of the square root with $\Im z>0$. I have to study its continuity in the set $$A=\lbrace z|a\leq\Re ...
0
votes
0answers
159 views

Give an example function that is solution for Dirichlet - problem in unitdisk D.

Give an example function that is solution for Dirichlet - problem in unitdisk D. I have tried border function $f:\partial D \rightarrow \mathbb{R}$, such that $f(z)=Re(e^{i\theta})$, but $\theta \in ...
2
votes
0answers
42 views

For compact $K \subset \mathbb C$, show that $u(z) = -\log(\mathrm{dist}(z,K))$ is subharmonic

Let $K \subset \mathbb C$ be compact, and let $u(z) = -\log(\mathrm{dist}(z,K))$ be defined on $\mathbb C \setminus K$. May I get a hint for proving that $u(z)$ is subharmonic? Subharmonic, here, is ...
2
votes
2answers
751 views

Order of the pole

What is the order of the pole at $z=0$: $$f(z)=\frac{1}{(2\cos(z)-2+z^2)^2}$$ and find and classify the isolated singularities of: $$\frac{1}{e^z-1}$$ My attempt: If I let ...
0
votes
1answer
1k views

Using Cauchy's integral formula to evaluate a function

This problem is from Brown/Churchill Complex Variables and Applications, 8th edition 2009. Section 52, exercise 2, subsection (a) How do I show that the integral of the function $g(z) = ...
2
votes
3answers
162 views

why can infinite product only be zero if one of the factors is zero?

I was reading about the Riemann zeta function in the region Re(Z) > 1, where it can be represented by the Euler product formula. And the book mentioned that there can be no zeros in this region, since ...
4
votes
1answer
176 views

Magnetic fields and the complex plane

The electrostatic potential $\varphi$ must satisfy Laplace's equation in regions without charge: $$\nabla^2 \varphi = 0.$$ If there is no $z$ dependence in the problem we are solving, we can choose ...
0
votes
1answer
151 views

Exponential sum identity

How do I show that $$\sum_{|j| \leq J} (J-|j|) e(j \alpha) = \left| \sum_{j=1}^J e(j \alpha) \right|^2,$$ where $e(n)=e^{2 \pi i n}$ and $\alpha \not \in \mathbb{Z}$? Thank you very much in advance! ...
0
votes
1answer
120 views

Why isn't this a counter example to "Linear Fractional Transformations are Automorphisms of $\mathbb C \cup \infty$

$\frac{Z-1}{2Z-4}$ is a linear fractional transformation, but it cannot take on the value $\frac{1}{2}$ -- so how can it be an Automorphisms of the Extended complex plane?
4
votes
1answer
146 views

change of variables in contour integration problem

On this answer: http://math.stackexchange.com/a/282675/65097, we see that $$\int_{-\infty}^{\infty} \: \frac{t^2}{t^4+1} dt = \int_0^{\infty} \frac{\sqrt{x}}{x^2+1} dx$$ from the change of ...
0
votes
2answers
85 views

How to show that $f-g$ is imaginary constant in $\mathbb{D}$?

How to show that $f-g$ is imaginary constant in $\mathbb{D}$? Let $f$ and $g$ be continuous functions in $\bar{\mathbb{D}}$ and analytic in $\mathbb{D}$. Show that if $\mathfrak{R}f=\mathfrak{R}g$ at ...
0
votes
1answer
123 views

Is $z^2-i(x^2-y^2)$ analytic in $\mathbb{D}$?

Is $f(z)=z^2-i(x^2-y^2)$ analytic in $\mathbb{D}$? I think it is because it is complex differentiable at $\mathbb{D}$. In other words it has unique complex derivative at all points of $\mathbb{D}$. ...
0
votes
1answer
104 views

Question on harmonic conjugates and Liouville's theorem

Suppose that $f(z)=u+iv$ is entire, and the harmonic function $u(x,y)$ has an upper bound. Then how to show that $u(x,y)$ must be constant throughout the plane?
0
votes
2answers
187 views

Question on application of Liouville's theorem

Let $f$ be an entire function such that $\mid f(z) \mid \leq A \mid z \mid$ for all $z$, where $A$ is a fixed number. Show that $f(z)=a_1z$, where $a_1$ is a complex constant.
2
votes
1answer
53 views

Is this estimation correct?

I have to estimate the following quantity $$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$ in $\mathbb{R}^3$ ($\lambda>0$) where ...
3
votes
2answers
172 views

A problem on Residue Theorem

Today I had a problem in my test which said Calculate $\int_C \dfrac{z}{z^2 + 1}$ where C is circle $|z+\dfrac{1}{z}|= 2$. Now, clearly this was a misprint since C is not a circle. I tried to find ...
3
votes
2answers
487 views

functions with multiple branch points

EDIT: My original question was poorly worded and thus confusing. So I'm going to edit it and then give a brief answer. $ $ Let $f(z) = \displaystyle \sqrt{1-z^{2}}= \sqrt{(1+z)(1-z)} = ...
1
vote
1answer
49 views

determining complex function

Problem Let $f$ be holomorphic in $D=\{|x+iy|< 1\}$, with $|f|\le |y|^{-1/2}$ $\lim_{r\rightarrow 1} f(re^{i\theta}) = 0$, for any $\theta\in[0,2\pi]$ Prove $f = 0$. It is an old qualify ...
2
votes
3answers
126 views

Cauchy Integral theorem

Let $f(z)=\sum^{\infty}_{k=0}\frac{k^3}{3^k}z^k$, compute $\int_{|z|=1}\frac{f(z)}{z^4}dz$ and $\int_{|z|=1}\frac{f(z)sinz}{z^2}dz$. I do not know how to do these problems. I know it is a ...
2
votes
2answers
138 views

Cauchy's Integral Formula

Please can someone help me understand how to use the cauchy's integral formula? I have put a picture of a question which i am struggling to get the correct answer for! I have the formula but i am a ...
6
votes
1answer
900 views

Complex roots of $z^6 + z^3 + 1 = 0$

The equation I'm trying to solve is $f(z) = 0$ where $$f(z) = z^6 + z^3 + 1$$ I already tried the following: randomly throwing in complex numbers and real numbers, rational root theorem, banging my ...
3
votes
2answers
199 views

Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
5
votes
2answers
378 views

Entire, $|f(z)|\le1+\sqrt{|z|}$ implies $f$ is constant

I am stuck on the following question. Given that $f$ is an entire function with $|f(z)|\le1+\sqrt{|z|}$ for all $z\in \mathbb{C}$, show that $f$ is constant. Can anyone give me a hint to get me ...
5
votes
3answers
73 views

Determine complex polynomial

Problem Let $P(z) = z^n + a_{n−1}z^{n−1} + \cdots + a_1z + a_0$ be a polynomial of degree $n > 0$. Show that if $\lvert P(z) \lvert \le 1$ whenever $\lvert z \rvert = 1$ then $P(z) = z^n$. I ...
3
votes
3answers
510 views

Analytic function f constant if $f(z) = 0$ or $f'(z) = 0$ for all $z$.

Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be analytic and suppose that for all $z \in \mathbb{C}$, at least one of $f(z)$ and $f'(z)$ is equal to 0. Proof that $f$ is constant. Any ideas? Thanks.
2
votes
1answer
1k views

Laplace transform of the Bessel function of the first kind

I can't figure out why my evaluation of $\displaystyle \int_{0}^{\infty} J_{n}(bx) e^{-ax} \ dx \ (a,b >0, \ n=0,1,2, \ldots)$ is off by a factor of $ \displaystyle \frac{1}{b}$. $$ \begin{align} ...
5
votes
4answers
369 views

Need help proving this integration

If $a>b>0$, prove that : $$\int_0^{2\pi} \frac{\sin^2\theta}{a+b\cos\theta}\ d\theta = \frac{2\pi}{b^2} \left(a-\sqrt{a^2-b^2} \right) $$
2
votes
0answers
96 views

Schwarz Lemma in Differential Form

Suppose $w=f(z)$ is a conformal self map of $\mathbb{D}$. From Schwarz Pick Lemma we have $|\frac{dw}{dz}|=\frac{1-|w|^2}{1-|z|^2}$. Could any one explain me In differential form how this becomes ...
4
votes
2answers
437 views

Show that $f(z)$ has no antiderivative in $\,S=\mathbb{C}\setminus \{-i,i\}$

$f(z)=\frac{1}{z^{2}+1}$ I know that you can do this using a proof by contradiction and by showing that if you assume it has an anti-derivative that it wouldn't follow the fundamental theorem of ...
0
votes
1answer
116 views

Sum of all the residues of the function $a(z)/b(z)$

Let $a(z)$ and $b(z)$ be polynomials such that $ \deg(b) \ge \deg(a)+2$. Find the sum of all the residues of the function $a(z)/b(z)$. In class, I learned that $$ - \text{ sum of all residues of ...
0
votes
2answers
154 views

Constructing Taylor series

I am having trouble constructing the answer to this problem, which is also linked here: Convergence Properties of the Taylor Series for $\frac{1+z}{1-z}$. Find and state the convergence properties ...
1
vote
2answers
74 views

Function analytic on a disk maps linear segments to line segments

Suppose f is analytic on a disk D and the image of every horizontal line segment is a horizontal line segment. Can I get some suggestions for an approach to proving that the derivative of f is ...
3
votes
3answers
74 views

Convergence of a sequence function

Show that the sequence of function $F_n(z)=\frac{z^n}{z^n-3^n},\ n=1,2,...,\ $ converges to zero for $|z|<3$ amd to $1$ for $|z|>3$. How can I show this? I can see why it will converge to ...
1
vote
2answers
184 views

Elementary proof of convergence

Prove that if the sequence $\{z_n\}_{n=1}^{\infty}$ converges, then $(z_n-z_{n-1})\to 0$ as $n\to\infty$. My attempt: Suppose $\text{lim}_{n\to\infty}z_n= L$, then $\exists$ $\epsilon >0$ ...
4
votes
1answer
162 views

Is there any specific formula for $\log{f(z)}$?

Let $f(z)$ be a nonvanishing analytic function on a simply connected region $\Omega$. Then there is an analytic function $g(z)$ such that $e^{g(z)}=f(z)$. Is there any specific formula for $g(z)$? ...
0
votes
2answers
106 views

Harmonic function, existence of a constant

May i ask you for a little help about a problem with harmonic function? It seems to be not that difficult, in a way even intiutively obvious but i don't really know how to show this explicitly. We ...
2
votes
3answers
382 views

Evaluation of definite integral using residue theorem

$$ \int^{+\infty}_{-\infty} \frac{x-1}{x^3-1} dx$$ I need to evaluate the above integral . My idea is to consider the same integral but with the $x$'s as $z$'s, over the complex plane, have a ...
0
votes
3answers
86 views

Differentials (infinitesimals) in complex analysis

We have a complex function $ w(z)=w(x+iy)$, and we can write $w(x,y)=u(x,y)+iv(x,y)$. The derivative is $$\frac{dw}{dz}=\frac{1}{2}(\frac{\partial z}{\partial x}-i\frac{\partial w}{\partial y})$$ ...
4
votes
1answer
1k views

Maximum Modulus Exercise

Using the maximum modulus theorem in complex analysis, what is a good technique for finding the maximum of $|f(z)|$ on $|z|\le 1$, when $f(z)=z^2-3z+2$? Got some really nice answers below, so I ...
16
votes
3answers
928 views

Definite Integral $\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$

I want to prove that $$\int_0^{\pi/2} \frac{\log(\cos x)}{x^2+\log^2(\cos x)}dx = \frac{\pi}{2}\left(1-\frac{1}{\log 2}\right)$$
3
votes
2answers
234 views

$f(z)= az$ if $f$ is analytic and $f(z_{1}+z_{2})=f(z_{1})+f(z_{2})$

If $f$ is an analytic function with $f(z_{1}+z_{2})=f(z_{1})+f(z_{2})$, how can we show that $f(z)= az$ where $a$ is a complex constant?
2
votes
2answers
68 views

Is there an analytic function applying formula?

Is there an analytic function $f$ in $\mathbb{C}\backslash \{0\}$ s.t. for every $z\ne0$: $$|f(z)|\ge\frac{1}{\sqrt{|z|}}\, ?$$
5
votes
1answer
106 views

Does a bounded convergent power series on an open disc extend to the boundary?

Here is my question: Suppose that $|\sum_{n=0}^{\infty}a_nz^n| \leq M$ for all $z \in D_r$ (the open disc or radius $r$). Does this power series converge on $\partial D_r$?