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, ...

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2
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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 ...
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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
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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 ...
2
votes
2answers
114 views
+50

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
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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
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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 ...
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1answer
50 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 ...
0
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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
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0answers
36 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
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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
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0answers
53 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
51 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.
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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
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1answer
37 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
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3answers
49 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
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2answers
39 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
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1answer
16 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
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1answer
33 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
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1answer
32 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 ...
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0answers
17 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 ...
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1answer
35 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
49 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
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0answers
42 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
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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
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0answers
168 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
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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 ...
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1answer
37 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
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1answer
34 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
116 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
75 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
36 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
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1answer
36 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
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1answer
35 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 ...
4
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0answers
42 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
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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 < ...
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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 ...
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2answers
42 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: ...
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0answers
28 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 ...
0
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1answer
25 views

How to express a contour

How would I express the contour which is the portion of the unit circle in the left hand plane going from i to -i. I though the contour would be $y(t)=e^{it}$ $t {\in} [-{\pi}/2,{\pi}/2]$ but this ...
0
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1answer
25 views

Contour integrals and Cauchy's theorem

1) Let $C$ be a contour beginning and ending at 1. Suppose that $f(z)$ is analytic on $C$. Then is it true that the contour integral of $f$ around $C$ is 0? This looks to be true by Cauchy's theorem ...
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0answers
12 views

Doubt on invariant points of bilinear/Möbius transformation

Here is what I read as the intro to "Invariant points of bilinear/Möbius transformation": If z maps into itself in the w-plane(i.e.,w=z), then $\displaystyle w=z=\frac{az+b}{cz+d}$ or ...
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1answer
133 views

Why does the Fund. Theorem of Contour Integrals Need Continuity?

Why does the Fundamental Theorem of Contour Integrals need continuity? When defining the integral in real analysis we don't require continuity of the function we are integrating, is it necessary to ...
0
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1answer
21 views

clarification of what does it mean a Hilbert lattice?

I was searching to find and understand the meaning of a Hilbert Lattice. I could not find on Google something simply define what is a Hilbert lattice. Can any one help me and explain to me what is a ...
1
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1answer
49 views

Closure of the set in complex analysis…

Here's the problem: In each case, sketch the closure of the set: (a) $−π < arg(z) < π$, $(z \neq 0)$ (b) $|Re (z)| < |z|$ (c) $Re(\frac{1}{z}) \leq \frac{1}{2}$ (d) ...
5
votes
1answer
75 views

To prove this complex polynomial has all zeros on unit circle

I'm trying to prove a self-inversive polynomial $P(z) = \sum\limits_{n=0}^{N-1}a_nz^n$ has all its roots on the unit circle. The coefficients are such that $ a_n = e^{j(n-\frac{N-1}{2})\pi u_0} - ...
0
votes
1answer
38 views

True or false logarithmic branches

Say whether the following are true or false. Give a short proof. 1) $log(-z)+i{\pi}$ is a branch of the logarithmic function whose branch cut is the non-negative real axis 2)If $g(z)$ is a branch of ...
3
votes
1answer
74 views

Find roots of $3z^{100} - e^z$ in the unit disc.

This question was given in an exam in complex analysis: Let $f \left( z \right) = 3z^{100} - e^z$. Find all of $f$'s roots in $D \left ( 0,1 \right)$ and show that they are simple roots. I've seen ...
3
votes
0answers
69 views

Infinitely many roots $z e^z = a, a\neq 0$

Spent some time trying to tackle this problem. It is supposed to use Rouche's Theorem, but not sure how. Show that $ze^z = a$ for $a \neq 0$ has infinitely many roots. Rouches: (1) $f$ and $g$ ...