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
1answer
13 views

Why $f(z)=z^2$ is single valued?

Why $f(z)=z^2$ is single valued where $z\in\mathbb{C}$? From definition we have $$z^2=e^{2 \log z}=e^{2(\ln|z|+i(2k\pi+Arg(z)))}$$ I dont get it ;/ Maybe it's getting late.
0
votes
0answers
9 views

Proper holomorphic maps and the degree of the map

Suppose f is holomorphic and maps U onto V, both being disks. If f is proper, does this induce a well defined degree for f? And does the converse hold? What are some tools that can help me see if ...
2
votes
0answers
12 views

Generalization of the Jacobi-Anger expansion to higher harmonics

I know the Jacobi-Anger expansion relation which gives the Fourier development of $e^{i z \cos(\theta)}$ and ${ e^{i z \sin(\theta)} }$, such that $$ \begin{cases} e^{i z \cos(\theta)} = ...
0
votes
0answers
8 views

Show that there is no branch of $z^{\frac{1}{n}}$ for $n \gt 1$ in the domain $B(0,1)-\{0\}$

Show that there is no branch of $z^{\frac{1}{n}}$ for $n \gt 1$ in the domain $B(0,1)-\{0\}$ Suppose there is one. Then Since $z^{\frac{1}{n}}=e^{\frac{1}{n}\log z}$, it is equivalent to saying that ...
1
vote
0answers
7 views

$\exists C \in \mathbb{C}^{\times}$, $n\in\mathbb{N}$: $f = C\Delta^n$

Let $f \in M_k(\Gamma)$ not null in $\mathbb{H}$. I want to show that there exists a $C \in \mathbb{C}^{\times}$ and $n\in\mathbb{N}$ with $f = C\Delta^n$. I think one can show that for a $k>0$ ...
1
vote
1answer
17 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
1
vote
0answers
25 views

Cuscs of a subgroup of $\Gamma$

I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to ...
1
vote
2answers
25 views

Find all singularities of a function and determine its types

Find all singularities of a function and determine its types $$f(z)=\frac{e^{iz}-1}{\sin{z}}e^{\frac{1}{z}}$$ I already showed, that $f$ has poles at points $z=\pi n$ where ...
1
vote
0answers
24 views

Transfomation for evaluation of real integrals using complex analysis [on hold]

In questions regarding calculation of improper integrals e.g $\int_{-\infty}^{\infty}\frac{x^2+1}{x^4+1} dx$, you consider the function $f(z) = \frac{z^2 + 1}{z^4+1}$ and apply the Cauchy's residue ...
1
vote
2answers
25 views

Points on a straight line (Complex Analysis)

I encouter a problem in complex analysis course : Let $a, b, $ and $c$ be three distinct points on a straight line with $b$ between $a$ and $c$. Show that $\frac{a-b}{c-b} \in \mathbb{R}_{<0}$. ...
2
votes
0answers
23 views

Cauchy-Riemann Equations Written as Complex Conjugate

Apparently, it can be shown that the Cauchy-Riemann equations can be written simply as, $df/dz^*=0$. I do not understand how it does not immediately follow from this that $df/dz=0$. When we proved ...
1
vote
0answers
11 views

Continuity of a function on the ring of formal power series, with a metric defined.

Let $E$ be the ring of formal power series over a field $K$. Consider $S,\ T \in E$. Define a metric $d$ on $E$ by $d(S,T)=0\ $ if $S=T$ and $a^{(-k)}$ for $k=\mathrm{order}(S-T)$, where $a>1$ is ...
1
vote
2answers
21 views

An explanation of how the range of a complex function works? Specifically $f(z)=z^2$ for$Re(z)>0$, $Im(z)>0$ and in the first quadrant..

I'm going through this complex analysis textbook, and it tells me that the range of the aforementioned function is $Im(w) \geq 0$. To me, that makes no sense. Could someone explain that, by chance? ...
1
vote
1answer
25 views

sketching lines and curves in the complex plane.

Well, my question is I a have the line equation x=1 and I need to know wich is thw image under $w=z^{2}$, then I parametrize it like $\theta=\pi /2$, the next step was to squred it $(\theta-\pi ...
1
vote
1answer
15 views

Series Expanding a Function: Complex Answer?

I have $$ f(x)=2\arccos\left(\frac{x}{2}\right)-x\sqrt{1-\frac{x^2}{4}} $$ and my friend says that the series of this function about $x=2$ (truncated to the first term) given $x\leq2$ is ...
0
votes
0answers
16 views

Coefficient in Taylor Series expansion [duplicate]

Find the coefficient of $(z-\pi)^2$ in the Taylor series expansion around $\pi$ if $$f(z) = \begin{cases} \frac{\sin z}{z-\pi} & \quad, z \neq \pi \\ -1 & \quad, z=\pi \end{cases}$$
2
votes
1answer
20 views

Power series of dependent and independent variables

Let $f(z,w)$ be an analytic function in two variables where $w=w(z)$ is dependent on $z$ ($z$ is the independent variable). Then $f(z,w)$ has a power series expansion centered at $(z_0,w(z_0))$ ...
1
vote
1answer
49 views

How to define Square Root

I'm trying to understand how to define the square root of a complex function "globally". Let's say we have some function from some set $X$ onto $\mathbb{C} - \{0\}$: $$ f:X\to\mathbb{C}-\{0\} $$ and ...
3
votes
3answers
49 views

Is the function complex differentiable at (0,0)?

(in Complex) $$ g(z) = \begin{cases} \frac{z^5}{|z|^4} & \text{if $z \neq 0$} \\ 0, & \text{if $z = 0$ } \end{cases} $$ For the function above, is it differentiable at $z=0$? I am trying to ...
2
votes
0answers
15 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
3
votes
1answer
41 views

Sketching curves in the complex plane

Well I really need your help here because I need to sketch the curve $|z-1|=1$ in the z-plane and then its image under the $w=z^{2}$ but the thing is that I dont know how to sketch that function. In ...
2
votes
0answers
14 views

Integrating around simple pole and semicircle

Let $f$ be a holomorphic function on $\mathbb{C}$ with simple pole at $z_0$. Then if $\Gamma$ is a circle around $z_0$ oriented counter-clockwise with radius $r$ and $r\rightarrow 0$, then ...
2
votes
1answer
30 views

Prove for some $z_0 \in C$ the function $f(z)=|z-z_0|$ is continuous on all of $\mathbb{C}$

Let $z_0\in\mathbb{C}$ and $f(z)=|z-z_0|$. Show that $f$ is continuous on $\mathbb{C}$. I expect to see a proof using the triangle inequality. Note a function $f$ is continuous on $\mathbb{C}$ if ...
1
vote
0answers
18 views

contour integral and limit: What is the condition of the interchange the order?

In real real analysis sense, the interchange between limit and integral is hold when integrand is uniformly converges. $i.e$ \begin{align} \int \lim f = \lim \int f \end{align} Here i want to ...
2
votes
2answers
36 views

Reflections of circles through a circle are circles

To make things easier, we will try to reflect some general circle through the unit circle. We can use the inverse of the Cayley transform to map our analytic arc in the $z$-plane to the real line in ...
2
votes
0answers
28 views

power series of $\frac{1}{1-z}$ at $ a$

Here is a problem and I'm confused whether I'm doing correct or not. Problem says, Write down a convergent power series at $a \neq 1$ which represents the function $\frac{1}{1-z}$ and find radius of ...
2
votes
1answer
16 views

Conformal transformation of complement of disk in upper half plane

Let $U$ be the complement in the half-plane $\operatorname{Im} z > 0$ of a disk of radius $a<1$ centered at $i$. I am looking for a conformal transformation that maps $U$ onto an annulus. Since ...
2
votes
3answers
35 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
1
vote
0answers
23 views

Cauchy's integral theorem and domain boundaries

On a homework assignment, I was asked if the following statement is true. If $f(z)$ is analytic in a simply connected domain $D$ and continuous in $\partial D$ then $\oint_{\partial D} f(z) = 0$. Is ...
1
vote
1answer
21 views

Contour Integration where Contour contains singularity

There are many theorems in complex analysis which tell us about integration $\int_{\gamma} f$ where $f$ is continuous (or even differentiable) in the interior of $\gamma$ except finitely many points. ...
2
votes
1answer
28 views

Simple complex analysis inverse

On page 113 of Churchill in explaining the $\arcsin{(-i)}$ it comes across $$ln(1-\sqrt{2})$$ which is fine but then it goes on to say that it is equal to $$ln{\frac{1}{1+\sqrt{2}}}$$ How do they ...
1
vote
2answers
44 views

Complex differentiability of $f(z)=|z|$

Why is the absolute value function $f : \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = |z|$ not complex differentiable at any point $z_0$ in the plane?
0
votes
1answer
28 views

Complex variables Open ball [on hold]

Let $f(z) = \frac1z$ be inversion. Given a real number $a$, let $R_a = \{z \in C : Im(z) < a\}$. Why is $f(R_a)$ an open disk, provided $a < 0$. What happens when $a \ge 0$?
5
votes
2answers
28 views

The Duplication Formula for the Gamma Function by logarithmic derivatives.

I was reading Ahlfors' "Complex Analysis" (second edition) and in Chapter 5, section 2.4, where he studies the Gamma Function, he proves Legendre's Duplication Formula: ...
0
votes
0answers
50 views

$\epsilon$-$\delta$ proof of a sinc limit in Complex variables [on hold]

I am stuck on the following problem : Prove (using $\epsilon$-$\delta$) that $$\lim_{z \rightarrow \pi/2} \frac{\sin z}{z} = 2/\pi$$ Basically I do not know how to get an estimate on ...
5
votes
3answers
50 views

taylor of $\frac{1}{z}$ at $a=-2$

I want to find the taylor series representation of $f(z)=\frac{1}{z}$ at $a=-2$. The point of this exercise is not to find some pattern in the derivatives, infact we are not meant to find any ...
1
vote
2answers
64 views

Inequalities involving the sine of Complex Variable z

Is there any relationship between $|\sin z|$ , $\sin |z|$ , and $|z|$ ??? I know in real variables for example we have that $|\sin x|\le|x|$
0
votes
1answer
29 views

Relationship of basis vectors of the complex plane

I am working on learning more about the connection of complex numbers and rotations in the context of rational geometry. Thanks ahead of time for any corrections on my best assertions. Let $B$ ...
0
votes
2answers
35 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
0
votes
1answer
60 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $a_0 , a_1 , a_2 , b_0 , b_1 , b_2 \in \mathbb{C} $ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( 1 + j b_0 ) ( 1 + j^2 b_0 ) ...
1
vote
4answers
36 views

Sum of nth roots of unity

Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$ Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} ...
1
vote
1answer
13 views

A question about analytic functions on the unit disk with $\Re[h(z)]=0$ and a double pole at $1$

Let $\bar{D}(0,1)$ denote the closed unit disk around $0$ and $D$ the unit circle. I am interested in obtaining a complex function, say h, with the following properties: $h(z)$ is analytic for ...
3
votes
2answers
36 views

Existence of a non-constant entire function

Which of the following statements are true? a. There exits a non-constant entire function which is bounded on the upper half plane $$H=\{z\in \mathbb C:Im(z)>0\}$$ b. There exits a ...
1
vote
2answers
40 views

What is the coefficient of $(z-\pi)^2$ in Taylor series expansion of $\sin (z)/ (z-\pi)$

I want to determine the coefficient of $(z- \pi)^2$ in Taylor series expansion of $f(z)=\sin (z)/ (z-\pi)$ if $z \neq \pi $, $-1$ if $z=\pi$ around $\pi$. How can this be done? I don't know how to do ...
2
votes
1answer
32 views

Conformal mapping of a semi-circle and a finite line

Can I map a semi-circle and a finite line separated by a distance $h$ to two parallel lines? Since I am new to con-formal mapping, I used the $w=atan(z$) con formal function but I guess this is for ...
1
vote
0answers
29 views

Find maximum of a function of a complex argument

I'd like to find the maximum of a (real) function of a complex argument. However, the function contains the $\Re(z)$ operator, so that the question is $$\operatorname{argmax}_z f(z,\overline{z})$$ ...
2
votes
0answers
24 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
-2
votes
3answers
50 views

Imaginary part of $ln(\sqrt{i})?$ [on hold]

Which of the following is the imaginary part of a possible value of $\ln(\sqrt{i})?$ (a) $\pi$ (b) $\pi/2$ (c) $\pi/4$ (d) $\pi/8$ I compute $\sqrt{i}=\dfrac{1+i}{\sqrt{2}}$, but how to proceed ...
1
vote
0answers
20 views

Infinite product representation for the Sine Integral $\mathrm{Si}(z)$

The infinite series representation of the sine integral (http://en.wikipedia.org/wiki/Trigonometric_integral, previous m.se question: Is there any infinite series representation of the sine ...
2
votes
1answer
25 views

Complex var. integral: $\oint_{|z|=1} \frac{z^2\ dz}{\sin^3{z}\cos{z}}$

Integrate $\displaystyle\oint_C \dfrac{z^2\ dz}{\sin^3{z}\cos{z}}$; $C \rightarrow |z|=1$ I already know that $|z|=1$ is a circumference with $r=1$ and center at $(0,0)$. I also know there are ...