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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Contour integral independant of parametrisation

I have a question about the definition of contour integrals in $\mathbb{C}$. The same question could be applied to line integrals in $\mathbb{R}^n$ though. $\Gamma \subseteq \mathbb{C}$ is called a ...
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29 views

Square root of several variables analytic function

The question is as follow: Let $H\subset \mathbb{C}^n$ be an simply-connected region. If $f$ is a nowhere vanishing analytic function on $H$, with $f(z)>0$ for all $z\in H\cap\mathbb{R}^n$, ...
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1answer
46 views

The identity $ \sqrt[n]{z}\sqrt[n]{w} = \sqrt[n]{zw}$ for complex numbers

In the general case, when $z$ and $w$ are two complex numbers, we have that $ (1) \sqrt[n]{z}\sqrt[n]{w} \neq \sqrt[n]{zw}$ For example, $\sqrt{-1}\sqrt{-1} \neq \sqrt{-1.-1} = 1$. However, there ...
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3answers
80 views

Mean Value Property to show that entire function is a constant

Let $f(z)$ be an entire function so that, $$ \int \frac{|f(z)|}{1 + |z|^3} dA(z) < \infty$$ where the integral is taken over the entire complex plane. Show that $f$ is a constant. I believe ...
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1answer
20 views

Holomorphic functions and complex conjugation

Suppose I have given two holomorphic functions $g,f:\mathbb{C}\backslash(-\infty,1]\rightarrow \mathbb{C}$ and I know that $\overline{ g(z)}=f(z)$ for all $\vert 2-z \vert <1.$ I am wondering if ...
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1answer
17 views

inequality by taking reciprocal or other way to check if pole lies inside unit circle

If $ a^2$ <1 is given in the problem then how do we prove that the pole z=1/a lies outside the unit circle ?
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16 views

Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
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1answer
154 views

Is it always true that $\partial f(U)=f(\partial U)$ when $f$ is holomorphic?

Let $D\subseteq\Bbb C$, $f:D\to\Bbb C$ holomorphic on $D$. Let $U$ be an open subset strictly contained in $D$: in this way $\partial U$ would be contained in $D$. So I was asking myself if $\...
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1answer
40 views

Sum over all residues is zero

Let $f$ be a rational function mapping to $\mathbb{C}$, $$f(z)=\frac{P(z)}{Q(z)}$$ with $\deg P\leq \deg Q -2$. I want to show that the sum over all residues is zero. What am I asked to show? I think:...
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2answers
43 views

Partial fraction decomposition of $\pi\cdot \tan(\pi z)$

Evaluate the partial fraction decomposition of $\pi \tan(\pi z)$ $$2\pi \tan(\pi z)=\cot\left(\frac{\pi}{2}-\pi z\right)-\cot\left(\frac{\pi}{2}+\pi z\right)$$ $$=\frac{2}{1-2z}+\sum_{k=1}^\infty \...
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1answer
73 views
+100

Sufficient condition for an holomorphic map to be conformal

Let $U,V\subseteq\Bbb C$ be open sets, let $f:U\to\Bbb C$ be holomorphic. If we want to prove that $f$ is a conformal map $U\to V$, my teacher said that is enough to check that $f$ is locally ...
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0answers
48 views

What's about the derivative of the Riemann zeta function?

The derivative of the Riemann Zeta function is $$\zeta'(s)=-\sum_{n=2}^\infty\frac{\log n}{n^s}$$ for $\Re s>1$. Question. Can you refers us in a short post, from a divulgative viewpoint (but ...
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0answers
16 views

Is there a $g$ defined on domain $D$ with $g' = f$, where $f$ is a holomorphic function on $D$? [closed]

Let $f$ be a holomorphic function on a domain $D$. Is there a $g$ defined on $D$ with $g' = f$?
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1answer
24 views

Exists holomorphic function on unit disk satisfying condition

Let $g: [0, 2\pi] \to \mathbb{R}$ be a continuous function with $g(0) = g(2\pi)$. Does there exist a holomorphic function on the unit disk such that for each $\theta \in [0, 2\pi]$, we have$$\lim_{r \...
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1answer
49 views

Radius of convergence of function strictly greater than $1$ or not?

Suppose $f$ is analytic on $\mathbb{D}$ with power series$$f(z) = \sum_{n = 0}^\infty a_n z^n.$$Let$$f_n(z) = a_0 + a_1z + \ldots + a_nz^n.$$Suppose that $f_n \to f$ uniformly on the closed unit disk. ...
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0answers
19 views

Entire function $g$ where $|g(z)| \le |z|^{3/2} + 1$ implies $g(z) = az$?

Suppose $g$ is an entire function with $g(0) = 0$ such that for all $z$,$$|g(z)| \le |z|^{3/2} + 1.$$Then does it necessarily follow that $g(z) = az$ for some $ a \in \mathbb{C}$?
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49 views

Identity theorem for $2\pi\mathrm i$ periodic function

Let $f$ be entire as well as real-valued along the lines $\operatorname{Im}(z)=0$ and $\operatorname{Im}(z)=\pi$. Show that $f$ is $2\pi\mathrm i$ periodic under these circumstances, that is $f(z+2\pi\...
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1answer
43 views

Maximum and minimum modulus principle

Let $U\subset \mathbb C$ be a bounded domain and $f:\overline{U}\to\mathbb C$ continuous and holomorphic $U$. Show that $|f(z)|\leq\max\{|f(w)|:w\in\partial U\}$ for all $z\in U$. Show that ...
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2answers
89 views

Show $\frac{\pi^2}{8}=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}$ [closed]

Knowing $$\frac{\pi^2}{sin^2(\pi z)}=\sum_{n=-\infty}^{\infty}\frac{1}{(z-n)^2}$$ how can I prove $$\frac{\pi^2}{8}=\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}\qquad?$$
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24 views

How to simplify the following expression involving Jacobian elliptic functions?

I would like to show that a certain elliptic function $F(x)$ (that is periodic, say with some period $h$) has exactly two zeroes in $[0,h)$. Let us recall some notation. Given a parameter $m \in [0,1]$...
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42 views

Function satisfying inequality has no root

Let $f$ be an entire function such that, for all $z \in \mathbb{C}$ with $|z| > 1$, $$ |f'(z)| < \frac{|f(z)|}{|z|^2} < 1 $$ Show that there is no $a \in \mathbb{C}$ such that $f(a) = 0$. ...
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Relationship between the residues $Res(g\circ\varphi,z_0)$ and $Res(g,w_0)$

Let $\varphi:U\rightarrow\mathbb{C}$ be holomorphic with $\varphi'(z_0)\neq 0$ for some $z_0\in U$. Let $g$ be another function having a pole of order $1$ in $w_0=\varphi(z_0)$. What is the ...
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0answers
11 views

Polynomial has not roots on disc D($0$,$1$) [duplicate]

Let p($z$)=$a_n$$z^n$+..+$a_0$ with 0< $a_n$ $\le$ $a_{n-1}$ $\le$...$\le $$a_0$ .Show that p($z$) has not roots on D={ ℂ $\exists$ $z$ : $|z|<1$ }. thanks
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40 views

Why is a absolutely and uniformly convergent series in the complex plane holomorphic?

Suppose I have some series $f(z) = \sum_{k = 0}^\infty a_n(z)$, with $a_i$ holomorphic on $\mathbf{H}$, that is absolutely convergent for all $z \in \mathbf{H}$ and uniformly convergent on compact ...
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0answers
31 views

Completeness of 'Hardy Space' $H^2(D)$

Define Hardy Space $H^2(D)$ as a space of holomorphic functions $f$ on unit open disc $D=\{z\in\mathbb{C}:|z|<1\}$ endowed with the norm $$ ||f||^2=\sup_{0<r<1} \int_0^{2\pi} |f(re^{i\...
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2answers
87 views

Evaluating $\int_c\frac{1}{\sin\frac{1}{z}}\text{d}z$ over $C= \{z\big\vert|z|=\frac{1}{5}\}$

Evaluate $$\int\limits_{|z|=\frac{1}{5}} \frac{1}{\sin\frac{1}{z}}\text{d}z$$ My attempt: I know that this function has non isolated singularity at $0$, and simple poles at $\frac{1}{n \pi}$. ...
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65 views

Geometric interpretation of a complex set

These usually aren't too bad but I had difficulties thinking of what the set $$\{z\in\mathbb{C}:|z+i|=2|z|\}$$ looks like in the complex plane. I got as far as $$|z+i|=2|z|\Rightarrow \sqrt{(z+i)(\...
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1answer
46 views

Application of Rouché's theorem

I'm reading Stein & Shakarchi's Complex Analysis II, and I have a question about the proof proposition 1.1, chapter 8. Here is the proposition: (assume $U, V \subset \mathbb{C}$ open) If $f:U\...
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1answer
48 views

Find on which $z=x+iy\in\mathbb{C}$ the function $f(z)=(\overline{z}+1)^3 - 3\overline{z}$ is differentiable

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
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1answer
34 views

Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
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1answer
45 views

Decomposition of Harmonic function into sum of holomorphic and anti-holomorphic function

How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\...
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0answers
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Katok book Exercise 2.8.3 [closed]

Prove that for any holomorphic function w wich is not a polynomial there exist a number $\lambda = exp2\pi i \alpha$, where $\alpha$ is irrational, such that the linearized equation (2.8.3) does not ...
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1answer
41 views

Find and classify isolated singularities of $f(z) = \frac{z}{1-e^{z^2}}$ and calculate residues on them

Obviously, all isolated singularities will be of the form $z=\sqrt{2 \pi i k}$ for $k \in \mathbb{Z}$ but I don't know how to classify. I tried expanding $\frac{1}{f} = \frac{1-e^{z^2}}{z}$ to $- \...
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0answers
10 views

Decomposition of the transformations of the Hilbert curve into real and imaginary parts

I have the set of transformations to generate the Hilbert curve. The complex representation is $\zeta_0 z=\frac{1}{2} \bar{z} i$ $\zeta_1 z=\frac{1}{2} z+ \frac{i}{2}$ $\zeta_2 z=\frac{1}{2}z + \...
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1answer
29 views

Discontinuous complex integral

I would like to compute the integral $$ \frac{1}{2\pi i}\int_{c-i\infty}^{c+\infty}\frac{x^{s}}{s(s-1)}ds $$ for $ x \geq 1$ and $ c > 0$. I know that it should be $ \sim x $ and that if $ s-1 $ ...
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1answer
56 views

2D Fourier transform of characteristic function of stripe on xy plane

Given a stripe $X$ on the xy-plane, namely $X\subset\mathbb{R}^2$, with $X=\{(x,y)\,|\; mx-\frac{1}{2}t \le y \le mx + \frac{1}{2}t$} and its "characteristic" function $$ f(x,y) = \begin{cases} 1, ...
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32 views

problem on schwarz reflection principle

let f be an entire function on $\mathbb C$.Let g(z)=$\overline{f(\bar z)}$. Then which of the following are true: (1)If f(z)$\in \mathbb R $ $\forall z\in \mathbb R$,then,f(z)=g(z) (2)if f(z)$\in \...
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1answer
50 views

What type of singularity does $\displaystyle \frac{z-1}{exp(\frac{2\pi i}{z})-1}$ have at $z=0$

My answer: Since $\infty$ is an essential singularity of $e^z \implies 0$ is an essential singularity of $e^{\frac{1}{z}}$. I am assuming that it is also true that $0$ is an essential singularity of ...
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1answer
36 views

Maximum principle and open mapping theorem

Let $f:B_2(0)\to\mathbb C$ be holomorphic with $f(1)=1$ and $f(-1)=-1$. Show that there is a $z\in B_2(0)$ and a $\varepsilon>0$ s.t. $f(z)=1-\varepsilon$. Is it reasonable to say that since $|f|$...
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problem on taylor's expansion [duplicate]

let f be an entire function.Suppose,for each a$\in \mathbb R$, $\exists$ at least one coefficient $c_n$ in it's expansion i.e, f(z)=$\sum_{n=0}^\infty c_n(z-a)^n$ which is zero. then,which of the ...
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0answers
48 views

Biholomorphic function such that $\phi(1+i)$ and $\phi'(1+i)=1/\sqrt{2}$

Let $S=\{z \in \mathbb{C}: 0 \lt Arg(z) \lt \pi/2\}$ and $\mathbb{D}=\{{z\in \mathbb{C}:|z|\lt 1}\}$ Find a biholomorphic function $\phi: S \rightarrow \mathbb{D}$ such that $\phi(1+i)=0$ and $\...
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1answer
50 views

Difference of functions inequality

In one book on complex analysis I see the following: But $f$ is continuous at the point $z$. Hence, for each positive number $\varepsilon$, a positive number $\delta$ exists such that $$\lvert f(...
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1answer
26 views

To determine the values of constants for which the function is analytic.

If $f(x+ iy) = x^3 +a x^2 y+b x y^2+c y^3$ is analytic in the complex plane only if : A. $a=3i$ , $b= -3$, $c=-i$. B. $a= 3$ , $b= -3i$ , $c=1$ C. $a=-3$ , $b=3i$ ,$c =-1$ D. $a= 3$ , $b= i$ , $...
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1answer
31 views

Determine the nature of the isolated singularity

Given that $f(z) = \frac{1}{z} Log(1+z) $ Here's what I did so far: Because Log is not continuous along the negative real axis, hence, $z = -1$ is not an isolated singularity. The only isolated ...
6
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1answer
104 views

Prove that a complex-valued entire function is identically zero.

Suppose $f$ is entire and $$\iint_\mathbb{C}|f(z)|^2dxdy < \infty$$Prove that $f\equiv 0.$ So far I have: Suppose $f$ is bounded. Then $f$ is constant by virtue of Liouville and so the ...
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1answer
69 views

What is the integral of $\exp(z)/(1-z)$ around a circle of radius $1$ centered at $z=1$?

I am trying to prepare for my first course in complex variables. I have tried to read my text and as many articles "on line" as I can find but I am very confused. I wrote $\exp(z)/(1-z)$ as a series ...
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0answers
32 views

Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$.

I am struggling with figuring out the details of proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor. Setting is as follows. Let $\Omega$ be a noncompact Riemann ...
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0answers
25 views

Calculating the residue with series that has negative powers

I want to calculate the residue of $f(z)=(z-sin(z))^{-1}=\frac{1}{z^3}+ \frac{3}{10z}+..$ at 0. We get that expansion using binomial theorem using provided $z$ small the residue of $f$ at $a$ is $\...
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1answer
42 views

Series expansion of $1/(1+z^2)$ about the point I

I am trying to find a series representation for the complex function: $1/(1+z^2)$. The text I am reading gives: $1/(1+z^2) = 1/((z+I)(z-I)) = -I/(2(z-I)) +1/4 - I(z-I)/8 - (z-I)^2/16 + ...$ I do not ...
2
votes
1answer
34 views

General method: show subset of $\mathbb{C}$ is connected

Consider the two sets $$ A = \{z \in \mathbb{C} : |z^2 - 3| < 1\}, ~~~~ B = \{z \in \mathbb{C} : |z^2 - 1| < 3 \} $$ $B$ is connected, while $A$ is not. However, I have no idea how to prove this....