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)

0
votes
3answers
48 views

Showing that $\frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right)$

I'm trying to show that $$ \frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right) $$ but am having trouble figuring out exactly how to approach the ...
1
vote
1answer
60 views

Why is the algebraic number a whole number.

Assume the function $f$, analytic on some domain has a non-essential singularity $a$. Define the algebraic order $h$ of $f$ at $a$ to be the real number such that $\lim_{z\to a}|z-a|^k|f(z)|=0$ for ...
0
votes
0answers
11 views

If an analytic function has an algebraic order $h$ at infinity then $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity

Assume infinity is not an essential singularity of the analytic function $f$. Then how is $h$, the algebraic order of $f$ such that $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity? p.s. ...
3
votes
3answers
59 views

Bound in Complex Analysis

Can someone direct me towards the right way to approach this problem? Show $$\displaystyle \left|\int_{|z|=R} \frac{Log{z}}{z^2} dz\right| \leq 2\sqrt{2}{\pi}\frac{\log{R}}{R},\; \text{ for } ...
0
votes
1answer
21 views

Images of Regions Under Cayley's Transformation

I'm working on the following problem for my complex analysis course: I can't seem to find Cayley's transformation anywhere in our textbook - could someone clarify to me what it is? I've done a ...
7
votes
1answer
134 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
3
votes
1answer
33 views

How does Ahlfors define derivative on a Riemann Surface?

I'm reading a passage in Ahlfors (3rd Edition page 298) and he says the following: He has previously defined $G_0(f)$ to be the connected component of any germ generated by $f$. Then he wants to ...
2
votes
2answers
66 views

Goursat's theorem and residue theorem understanding

It seems to me that Goursat's theorem doesn't align with the residue formula, because with the residue formula we end up with a number different than zero. Could you help me find what I understood ...
1
vote
0answers
12 views

some questions on the soluction of the Dirichlet's problem in the unit disk

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
2
votes
3answers
48 views

Parametrizing curve for complex analysis integral

I'm trying to show that $$ \int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases} $$ Here's my attempt at a solution: We parametrize the curve at ...
0
votes
1answer
35 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
0
votes
1answer
23 views

Isometries of the plane and fixed lines

I am given that for all reflections $g$ there are infinitely many lines $L$ satisfying $g(L) = L$ which makes perfect sense (just take lines perpendicular to the axis of reflection). I am asked to ...
0
votes
0answers
33 views

Analytic continuation of a function

Let $$f(z) = A_0 + A_1(z-a) + A_2(z-a)^2 + \cdots$$ converge in some disk $|z - a| < r$. Following Weyl, we magically re-arrange this power series at point $b$ in this disk and the power series ...
2
votes
4answers
58 views

Showing that Gaussians are eigenfunctions of the Fourier transform

I'm having a bit of trouble on this problem: I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar ...
1
vote
1answer
61 views

$|f(z)| ≤ 16$ for $|z| = 4.$ Prove that $|f(3i)| ≤ 9.$

(a) Suppose that $f(z)$ is analytic for $1 ≤ |z| ≤ 4.$ Assume that $|f(z)| ≤ 1$ for $|z| = 1$ and $|f(z)| ≤ 16$ for $|z| = 4.$ Prove that $|f(3i)| ≤ 9.$ (b) Prove that there is no non-constant ...
0
votes
1answer
36 views

functional equation of entire functions shall have only constant solutions

Given an entire function $f$ with $f'(0)=0$ and a function $g$ holomorphic (at least) in $\mathbb D:=\{z\in\mathbb C\ |\ |z|<1\}$ such that $f*g$ is entire as well and satisfies the functional ...
1
vote
0answers
42 views

Show that $f(z)= \sum_{n=0}^{\infty} z^n$ is analytic in $|z| <R$ [closed]

Let $f(z)= \sum_{n=0}^{\infty} z^n$ with $|z| <R$ where $R$ is the radius of convergence of $f$. Then show that $f$ is analytic in $|z|<R$.
0
votes
2answers
26 views

Cauchy integral formula or something else?

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$ with $C_1(0):\left|z\right|<1$ positive. Additionally ...
2
votes
0answers
35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
1
vote
0answers
21 views

Properties of a specific Complex function

Consider a map $f_{p,q}$ from $\mathbb{C}^2$ to $\mathbb{C}$ is defined as $f_{p.q}(z,w)=\frac{p+q.z}{1+w}$ where the $p$ and $q$ are two complex numbers. What can we talk about continuity, ...
3
votes
0answers
52 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
0
votes
2answers
48 views

$n$-to-$1$ near zero of holomorphic function

Can someone explain to me why a holomorphic function that grows like a polynomial of degree $n$ is $n$-to-$1$ near it's roots? I keep reading this fact on this site, but I can't find an explanation.
0
votes
0answers
24 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
6
votes
1answer
34 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
1
vote
3answers
48 views

integrating $\int_{\gamma}e^zdz$ with $\gamma$ is the arc on the unit circle that unites one with i

I am stuck integrating $$\int_{\gamma}e^zdz$$ with $\gamma$ is the arc on the unit circle that unites one with i. I tried this : The integrand $\mathrm{e}^z$ is holomorphic for $\vert z \vert \le ...
2
votes
2answers
38 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
0
votes
1answer
15 views

Let $A = \{1/2 < |z| < 2\}.$ Is there an analytic function $f$ on $\mathbb{C} \setminus \{0\}$ so that $Im(f) < −1$ on $∂A$ and $f(1) = 0$?

Let $A = \{1/2 < |z| < 2\}.$ Is there an analytic function $f$ on $\mathbb{C} \setminus \{0\}$ so that the imaginary part $Im(f) < −1$ on $∂A$ and $f(1) = 0$? Explain your answer. I am ...
0
votes
1answer
32 views

Find a conformal map from $\mathbb{D}=\{z;0<\operatorname{arg} z<2π\}$ to $Ω=\{w;0<\operatorname{Im} w<π\}$.

Find a conformal map from $\mathbb{D}=\{z;0<\operatorname{arg} z<2π\}$ to $Ω=\{w;0<\operatorname{Im} w<π\}$. I am having difficulty with this question. Some help would be awesome. ...
3
votes
1answer
29 views

conformal mapping onto right half plane

Find a conformal map of $D:=\{z\in\mathbb{C}:|z-i|<\sqrt{2}$ and $|z+i|<\sqrt{2}\}$ onto the right half plane. My idea was to use $$f(z)=\frac{z+\sqrt{\sqrt{2}-1}}{z-\sqrt{\sqrt{2}-1}}$$ To ...
0
votes
0answers
16 views

Writing a Mobius transformation as two fonctions belonging to a specific set

I had to prove something about the following set of maps: $$ H \quad = \quad \{ z \ \mapsto \ \frac{\rho^2}{\bar{z}-m} + m \ : \ m, \rho \in \mathbb{R} \} \quad \cup \quad \{z \mapsto -\bar{z} +2 ...
0
votes
1answer
34 views

Holomorphic functions and Laplace's equation.

My book says that for any holomorphic function $f(z)=u(x,y)+iv(x,y)$, $u$ and $v$ satisfy Laplace's equation. $f$ is holomorphic $\implies$ * $u_x=v_y$ and $u_y=-v_x$, so ...
5
votes
1answer
47 views

Composition of an analytic function with a continuous function that is analytic

If $f$ is a continuous function such that $g(z)=\sin{f(z)}$ is analytic, then is $f$ analytic? I know we can take $f(z)=\bar{z}$ then $f$ is continuous but $g$ is not analytic. Same holds if we take ...
0
votes
0answers
41 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
1
vote
1answer
29 views

Order of a zero of a complex polynomial

Is there a quick and easy way to determine an order of a zero $z_0$ of a complex polynomial without having to derive it $n$ times and check if $\;f^{(n)}(z_0)=0$ or not, which requires a lot of ...
9
votes
0answers
152 views

The closed form of $\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$

What tools, ways would you propose for getting the closed form of this integral? $$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
1
vote
1answer
26 views

On Stein manifolds and constant functions

Stein manifolds are defined here: http://en.wikipedia.org/wiki/Stein_manifold#Definition Obviously, M is Stein implies that there is a non-constant holomorphic function defined in it. Is the converse ...
1
vote
1answer
36 views

Confused by the task given (involves identical inequality of functions)

The task says: Show that if some function $\;f(z)={1\over g(z)}$, where $g\not\equiv0$ is an entirely analytic function, then the isolated singularities of $\;f$ are exactly zeros of $g$ ...
1
vote
1answer
32 views

Identity theorem for polynomials in several variables

Let us assume that we are given two polynomials $f,g$ with real coefficients in several variables, say $x_1, \ldots, x_n \in \mathbb{R}$. Further, assume that $f_{|X} \equiv g_{|X}$, with $X$ being ...
2
votes
3answers
108 views

Integration by Euler's formula

How do you integrate the following by using Euler's formula, without using integration by parts? $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}$$ I did integrate it by parts, by ...
3
votes
1answer
73 views

Is there an analytic function $f : \mathbb{D} → \mathbb{D}$ with $f(0) = 1/2$ and $f′(0) = 3/4?$

(a) Let $\mathbb{D}$ denote the unit disk. Is there an analytic function $f : \mathbb{D} → \mathbb{D}$ with $f(0) = 1/2$ and $f′(0) = 3/4?$ Either find such a function $f$ or explain why it does not ...
1
vote
1answer
35 views

line integral explanation

I asked this on the calculus tag but I didn't get any good answers so I decided to ask it here. It is actually is related to complex analysis because I need to understand the line integral before I ...
1
vote
1answer
35 views

Question regarding singularity of a complex function

Consider the function $$f(z) = {1 \over (z-i)(z+i)}$$ with a Laurent series expansion at $z_0=i$ on a domain $\;\Omega=\left\{z\in \mathbb{C}:2\lt\left|z-i\right|\right\}$ $$\begin{eqnarray}f(z)={1 ...
1
vote
1answer
34 views

Poles of complex function…

The function is: $f(z) = \frac{1 - e^z}{z^4sin(1 + z)} $ I know that 0 and the points $n\pi - 1$ for an integer n are singularities. I want to calculate the order of this singularities. In this case ...
2
votes
1answer
20 views

Application of Rouché: Equality of a power series and a finite series

Let $f(z) = \sum_0^\infty{a_n z_0^n}$ be a complex power series with radius of convergence $R>0$ and let $z_0 \epsilon \, \mathcal{U}_R(0)$ an arbitrary point. I need to show with $Rouché$ : For ...
3
votes
0answers
39 views

Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
0
votes
1answer
18 views

Proving function has simple pole and residue

Suppose $f$ is analytic and not constant on the domain $D \subseteq \mathbb{C}$. If $z_0 \in D$ is a zero of $f$ of order $k$, show that the function $\frac{f'(z)}{f(z)}$ has a simple pole ...
0
votes
1answer
21 views

Laurent series of a complex function

I have 2 functions. I have to express the function in terms of a Laurent series. The first function is $f(z) = \frac{z^5}{z - 1}$ in the point $z_0 = 1$ for $1 < \parallel z \parallel < ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
1
vote
2answers
41 views

Cauchy residue formula

Calculate the integral of $1/z$ around $C$, where $C$ is any contour going from $-\sqrt{3}+i$ to $-\sqrt{3}-i$ and is contained in the set of complex numbers whose real part is negative. My answer: ...
0
votes
0answers
24 views

Maximum modulus principle, 3 questions

I have several questions regarding the maximum modulus principle, but first let me interpret my understanding of this theorem: Assuming we have some analytic, non-constant function ...