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)

5
votes
1answer
482 views

Properties of the Mandelbrot set

Are there any properties of the Mandelbrot set that can be analysed without a knowledge of complicated topology? Considering the fact that the set is based on a quadratic function, are there any ...
5
votes
1answer
197 views

Prove that the only root of the equation $z-\sin(z)$ in the unit disk is zero.

Prove that the only root of the equation $z-\sin(z)$ in the unit disk is $z=0$. My first thought is Rouche's Theorem, but I don't know any bounds on $|\sin(z)|$. Suggestions?
2
votes
1answer
80 views

we need to pick out the cases where $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic but not neccessarily constant.

we need to pick out the cases where $f:\mathbb{C}\rightarrow\mathbb{C}$ is analytic but not neccessarily constant. $1.$$ \Im(f'(z))>0$ for all $z$ $2.$ $f(n)=3\forall n\in\mathbb{Z}$ $3$ ...
1
vote
1answer
248 views

any two simply connected open set in the plane R^2 are diffeomorphic

Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
1
vote
1answer
127 views

Holomorphic function with uncountable set of zeros?

I am aware that on a region, this is only possible if the function is identical to zero. If the domain is not a region, is it possible to have a non-trivial holomorphic function with uncountable zero ...
2
votes
0answers
127 views

Show Smoothness by Morera

I'm trying to show smoothness on $(0,\infty)(\Re)$ of the following function: $$ f(t,x)=\sum_{n=-\infty}^\infty e^{-\large \frac{(x-2\pi n)^2}{2t}}\frac{1}{\sqrt{2\pi t}} $$ The function is ...
5
votes
2answers
120 views

Computing $\lim_{s \to 1} \Gamma \left(\frac{1-s}{2}\right) (s-1)$

I want to evaluate the following limit: $$\lim_{s \to 1}\; \Gamma \left( \frac{1-s}{2} \right) (s-1).$$ I know that the gamma function has simple poles at $-n$ for $n \in \mathbb{N}_0$ with residue ...
2
votes
2answers
179 views

$f^2+2f+1$ is a polynomial implies that $f$ is a polynomial

This is a complex analysis problem. Let $f$ be an entire function and $f^2+2f+1$ be a polynomial. Prove that $f$ is a polynomial.
1
vote
0answers
89 views

Modular form weight 0

Why is an entire modular form of weight 0 must be a constant? In particular, does a function defined on the upper half plane that is analytic everywhere, including i/infty, imply boundedness? Can we ...
2
votes
3answers
162 views

Prove inequality in complex numbers in an unit circle

Given $|\omega| < 1$, $\omega \neq 0$ and $|z| < 1$. Prove inequality: $$\frac{|\frac{|\omega|}{\omega}z+1|}{|1-z \bar \omega|} \le \frac{2}{1-|z|}$$ It is simple but i have problems with it. ...
3
votes
1answer
111 views

Domain of bijectivity of function $f:\mathbb{C}\rightarrow\mathbb{C}$

There is a type of problems in my course in Complex analysis that I don't fully understand them. Given function $f:\mathbb{C}\rightarrow\mathbb{C}$, $f(z)=z^2$. You must specify the analytic and ...
1
vote
3answers
154 views

Integrate $\int_{C} \frac{1}{r-\bar{z}}dz$ - conflicting answers

In an homework exercise, we're asked to integrate $\int_{C} \frac{1}{k-\bar{z}}dz$ where C is some circle that doesn't pass through $k$. I tried solving this question through two different ...
2
votes
1answer
62 views

Prove, that $f: S_{k} \rightarrow \mathbb{C}-\{0\}$ is a surjection.

For $k>0$ define $S_{k} := \{z=x+iy\in\mathbb{C}\mid |z|<k,\ \ \ \ k\cdot y>|x|\}\subset\mathbb{C}$ Let $f(z)=\exp(1/z)\ \ \ \text{for}\ \ \ z\neq 0$ Prove, that $f: S_{k} \rightarrow ...
2
votes
1answer
130 views

What is wrong in this solution for finding the $n-th$ derivative of $f(z)=\frac{1}{1+z}$ at the point $z_{0}=0$?

I was given a homework question to calculate $f^{(n)}(0)$ where $f(z)=\frac{1}{1+z}$. I now have a way to solve the question, but I don't understand why another way I tried gives a different and ...
1
vote
1answer
60 views

A problem on complex integration where $\gamma$ is a closed and continuously differentiable path in the upper half plane

Let $\gamma$ be a closed and continuously differentiable path in the upper half plane $\{z \in \mathbb{C} : z = x + iy,\; x, y \in\mathbb{ R}, \;y > 0 \}$ not passing through the point $i$. ...
0
votes
1answer
201 views

Change of argument of $\exp(z)-z$ on each side of a square

Show that as the positive integer $N$ tends to $\infty$, the change in argument of $e^z − z$ is bounded on $3$ sides of the square with corners $ \pm 2\pi N$ $\pm 2\pi iN$ but is unbounded on the ...
1
vote
1answer
191 views

if $f$ is entire then show that $f(z)f(1/z)$ is also entire

This is again for an old exam. Let $f$ be an entire function, show that f(z)f(1/z) is entire. How do I go about showing the above. Do I use the definition of analyticity?., Call g: f(z)f(1/z) and ...
4
votes
1answer
168 views

If $\,f(z)=\exp((z+1)/(z-1))\,$ then all singular points of $1/(f(z)-a)$ are simple poles

Here is a question from an old qualifying exam. Let $f(z)=e^{\frac{z+1}{z-1}}$. Show that $f$ maps the unit disc $D$ in to the unit disk.(I can show this using properties of LFT. Let ...
2
votes
1answer
211 views

An application of Rouche's theorem involving $e^z$

Put $p_n(z)=\sum_{k=0}^n\frac{z^k}{k!}$. Show that for any $r>0$ and any $n\ge 0$, there exists a point $z_0$ with $|z_0|=r$ such that $|p_n(z_0)|=|e^{z_0}|$. This is actually the second part ...
3
votes
0answers
91 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
25
votes
6answers
2k views

Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis

I am studying for an exam and I have been studying my butt off during the winter break for it. During the course of my study I have written down quite a number of tricks, which in my opinion were ...
2
votes
1answer
175 views

How to calculate $\oint_{C}\frac{dz}{z(z-1)(z-2)}$ when $C$ is a circle around the origin with radius $1.5$?

I wish to calculate $\oint_{C}\frac{dz}{z(z-1)(z-2)}$ when $C$ is a circle around the origin with radius $1.5$. I guess that I should somehow apply Cauchy's integral formula here, but ...
3
votes
1answer
78 views

Characteristic function as limit of integrals.

I've seen the following claim in a book without proof and don't know why it holds. Let $a<b \in\mathbb{R}$. Then the integral $$\frac{1}{2\pi ...
4
votes
1answer
167 views

on the convergence exponent of zeros of entire functions

Let $\{z_j\}$ be the sequence of zeros on an entire function $f$. We define the convergence exponent of $\{z_j\}$ as $$b=\inf\left\{\lambda>0\ \text{s.t.}\ ...
1
vote
2answers
107 views

Quotient of entire functions which is also entire.

Suppose $f,g$ are entire functions, such that $\frac{f}{g}$ is also entire. Can I conclude that the set of zeros of $g$ is contained in the set of zeros of $f$? I think the answer is yes, otherwise ...
2
votes
1answer
120 views

Value of Arg(f(z)) after rotation of z

An elementary problem asks for Arg($f(z)$) after a single complete counter-clockwise rotation of the point z about the origin, beginning at the point z = 2 and taking the angle there to be 0, with ...
2
votes
1answer
65 views

What can you say about $f(z)$ with finitely many poles?

Assume that $f$ is analytic with finitely many poles {$z_1,z_2,...,z_n$}. At $z=z_i$, $f$ has a pole with multiplicy $m_j>0$. Suppose that $|f(z)\le C(1+|z|)^m$ for $|z|>R$. What can you say ...
1
vote
4answers
90 views

Minimum value of $|z-w|$ where $z,w \in \mathbb C$ such that $|z|=11$, and $|w+4+3i|=5$?

I was thinking about the problem: What is the minimum value of $|z-w|$ where $z,w \in \mathbb C$ such that $|z|=11$, and $|w+4+3i|=5$? My attempts: I notice that $|z-w| \geq |z|-|w|=11-|w|$. Also if ...
5
votes
1answer
257 views

Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear?

This is an exercise in complex analysis: Let $\Omega\subset{\Bbb C}$ be open and bounded, and $\varphi:\Omega\to\Omega$ a holomorphic function. Prove that if there exists a point $z_0\in\Omega$ ...
0
votes
1answer
110 views

sign of roots of a quadratic equation with complex coefficients.

Consider $x^2+ax+b=0$, where $x$ is the variable and $a,b$ are complex coefficients. Is there any condition on $a$ and $b$ which makes sure the roots of the equation have negative real parts?
2
votes
1answer
121 views

Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, then$f (i) $ is

Let $f (z) $ be an entire function such that $|f (z)|≤K|z|$, $∀z∈\mathbb{C}$, for some $K>0$. If $f (1) =i$, the value of $f (i) $ is (A) $ 1 $ (B)$-1$ (C) $i$ (D) $-i$ how can I able to ...
0
votes
1answer
62 views

order of growth of a counting function

Let $\{a_n\}$ be a sequence of complex numbers. For every $t>0$, define $n(t)=$ the number of $a_n$ satisfying the inequality $|a_n|\leq t$. We call $n(r)$ the counting function for the sequence. ...
2
votes
1answer
326 views

Evaluate a double infinite summation

$$ H=\sum_m\sum_n'\frac{1}{(m-1+nz)(m+nz)} $$ The summation of $m,n$ is over $\mathbb{Z}$, and skips $(1,0),(0,0)$,and $z\in\mathbb{H}$ (The upper half plane). The series comes from Jean-Pierre ...
0
votes
1answer
47 views

A Triangular Domain with Different Bounds on the Sides

I have the following problem: Let $K$ be the equilateral triangle centered at 0. Assume that $f$ is continuous on $K$ and analytic inside of $K$, and assume that $|f(z)|\leq 8$ on one of the sides ...
0
votes
1answer
161 views

Exercise complex variable, series.

Get the power series expansion centered at the origin of the function f, and calculate the radius of convergence of the corresponding series in each of the following cases: ...
2
votes
3answers
129 views

$e^{i\theta_n}\to e^{i\theta}\implies \theta_n\to\theta$

How to show $e^{i\theta_n}\to e^{i\theta}\implies \theta_n\to\theta$ for $-\pi<\theta_n,\theta<\pi.$ I'm completely stuck in it. Please help.
0
votes
0answers
66 views

In my Möbius transformation, what elements map to what?

Write down the subgroup $M_{\{0, 1 + i, \infty\}} \subset M$ preserving the set $\{0, 1 + i, \infty\}$ together with the explicit isomorphism $S_3$. I thought the transformation we need is the ...
1
vote
1answer
146 views

Complex Variable, properties.

Let $ f $ be a non-constant entire function. Justify that $ f $ satisfies one of the following two statements: (a) For all $ w \in \mathbb{C} $, the equation $ f(z) = w $ has a solution. (b) For all ...
0
votes
2answers
37 views

Complex Variable. Linearly dependent. [duplicate]

Possible Duplicate: Holomorphic functions and limits of a sequence Let $\Omega$ a domain and $f,g$ holomorphic function in $\Omega$. Suposse that $\exists$ a sequence $\{a_n\}$ in $\Omega$ ...
1
vote
2answers
43 views

Proving that a holomorphic $f$ such that $\lim_{z\rightarrow0}f(z)=\lim_{z\rightarrow\infty}f(z)=\infty$ has a zero.

Let $f\in H(C\backslash\{0\})$ a function such that $$\lim_{z\rightarrow0}f(z)=\lim_{z\rightarrow\infty}f(z)=\infty.$$ Prove that $f$ have some zero in $C\backslash\{0\}$.
1
vote
1answer
106 views

Entire function. Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$

Let $f$ a entire function: $f(R)\subset R.\;$ Prove that $f(\bar{z})=\overline{f(z)}, \forall z\in C$
1
vote
3answers
361 views

Prove that $\sum z^n/(1-z^n)$ is holomorphic on the unit disc.

Prove that the series $$f(z)=\sum_{n>1}\frac{z^n}{1-z^n}$$ converges in the unit disc $D=\{z:|z|<1\}$ and defines there a holomorphic function.
1
vote
2answers
278 views

Entire function with prescribed zeros

I want to construct an entire function which vanishes at points $n+in$ for all $n$ integers. I'm looking for the most simple entire function which satisfies this condition, in the sense that the ...
4
votes
1answer
374 views

order of growth of entire function

Show that $$f(z)=\frac{\sin\sqrt z}{\sqrt z}$$ is an entire function of finite order $\rho$ and determine $\rho$. I observed that the two determinations of the square root differ only for the signum. ...
4
votes
1answer
717 views

Argument principle: number of zeroes of $f(z)=\cos(z)-1 +z^2/2$ in the unit disk

I am trying to work on this old qual exam. Here is the question: Find the number of roots (counting multiplicities) of the function $$f(z)=\cos(z)-1 + \frac{z^2}{2}$$ inside the domain $\vert ...
0
votes
1answer
404 views

Must a meromorphic function on a compact set have same number of zeros and poles?

Let $f:X\rightarrow\mathbb{C}\cup\{\infty\}$ be a meromorphic function while $X$ is compact. Must $f$ have same number of zeros and poles?
3
votes
2answers
68 views

Why function $j(\tau)$ has degree 1?

We have $$ j(\tau)=\frac{1}{q}+\sum_{n=0}^{\infty}a_nq^n, a_n\in\mathbb{Z},q=e^{2\pi i\tau} $$ Then it is said that because $j$'s only pole is simple, $j$ has degree 1 as a map ...
1
vote
0answers
112 views

Proving that $ e^z = z+\lambda$ has exactly $m+n$ solutions $z$ such that $-2\pi m<\Im z<2\pi n$

I need to prove that for $\lambda\in \mathbb{C}$ and for $m,n\in\mathbb{Z}$ large enough, the equation: $$ e^z = z+\lambda$$ has exactly $m+n$ solutions $z$ such that $-2\pi m<\Im z<2\pi n$, ...
2
votes
1answer
100 views

Residue Calculus Integral computation

I ran into this problem when I was doing some residue computations. For real $a\neq0$, compute, $$I=\int_{-\infty}^{+\infty} \frac{e^{iax}}{(x+i)^3} $$ Be sure to treat both cases when $a<0, ...
0
votes
1answer
145 views

Verify Casorati-Weierstrass on example

How can I verify Casorati-Weierstrass theorem on the example ?$$f(z)=\sin\frac{1}{z}$$