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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1answer
19 views

Showing $f(z)=2xy+i(x^2+y^2)$ is defined on all of $\Bbb C$

What does it mean to be defined on all of $\Bbb C$? That is has no points at infinity? How do I show the below is defined on all of $\Bbb C$? $$f(z)=2xy+i(x^2+y^2)$$ Is it something to do with ...
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1answer
43 views

$\lim_{z\to 0} \frac{z}{\overline{z}}\text { does not exist }$

How can I make this rigorous? $$\lim_{z\to 0} \frac{z}{\overline{z}}\text { does not exist }$$ Proof: $$\lim_{z\to0}\frac{x+iy}{x-iy} \text{ taking } y\ne 0, x\to 0 \implies ...
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0answers
6 views

Prove that for every $g$ meromorphic there exists an entire function $f$ such that $f(z)\neq g(z)$ for all $z$ in $\mathbb C $

Prove that for every $g$ meromorphic there exists an entire function $f$ such that $f(z)\neq g(z)$ for all $z$ in $\mathbb C $. This problem is in pg 137 in Classical Topics in Complex function ...
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1answer
14 views

$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$

$$\lim_{z\to\infty} \frac{(az+b)^2}{(cz+d)^2}=\frac{a^2}{c^2} \text{ if }c\ne0$$ Now I am not sure how to prove this. Can I ignore the pesky square and do this? $$\lim_{z\to\infty} ...
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2answers
27 views

How find $\max _{z: \ |z|=1} \ f \left( z \right)$ for $f \left( z \right) = |z^3 - z +2|$

Let $f : C \mapsto R $, $f \left( z \right) = |z^3 - z +2|$. How find $\max _{z: \ |z|=1} \ f \left( z \right)$ ?
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8 views

Explain the geometrical interpretation a pair of harmonic function conjugated each other.

Explain the geometrical interpretation a pair of harmonic function conjugated each other. Could you help me? I am wondering how to draw it but unfortunately my abstract imagination can't cope with ...
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3answers
49 views

How to finish proof of $ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$

I'm trying to prove the identity $$ {1 \over 2}+ \sum_{k=1}^n \cos(k\varphi ) = {\sin({n+1 \over 2}\varphi)\over 2 \sin {\varphi \over 2}}$$ What I've done so far: From geometric series ...
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2answers
18 views

Solutions of an exponential function

Find all solutions of $e^{z} = -1+i$ These are the things for what I did: 1) Let $z=x+iy$ 2) $e^{iy} = e^{z} = √2(e^{i(3π/4+2kπ)})$ 3) Equate moduli and arguments to see that: ...
3
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1answer
19 views

What's the $z$-derivative of $|g|^2$ for $g(z)$ analytic?

Let $g\colon \mathbb{C} \to \mathbb{C}$ be holomorphic in a domain. What's $$\frac{\partial}{\partial z} (g \bar{g}).$$ I would think that since $\bar{g}$ is independent of $z$ (it's only dependent on ...
4
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1answer
45 views

How can I maintain notes while self studying Maths?

Thank you for stopping by this thread. I'm an engineering student rekindling an interest in Maths. I just love studying Maths in my free time (and sometimes it trespasses into my non free time). I ...
0
votes
1answer
26 views

Factorization $\cos(z) - \sin(z)$

How do I find the product expansion of $\cos z - \sin z$ We have $\cos z = \sin z$ iff $z = \pi/4 + k \pi$ where $k$ is an integer. The sequence $\sum (r/(|\pi/4 + k \pi|)^2$ converges For some ...
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0answers
11 views

Limit of sequence of analytic functions

If $\Omega_1$ and $\Omega_2$ are two nonempty disjoint open subsets ${\bf C}$ and $\{f_n\}$ is a sequence of analytic functions from $\Omega_1 \to \Omega_2$ which converges pointwise to a function $f ...
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0answers
25 views

$\frac{1}{2i\pi} \int{\frac{1}{(z^{2}-a^{2})^{1/2}}dz}$

Can you please integrate $$\frac{1}{2i\pi} \int{\frac{1}{(z^{2}-a^{2})^{1/2}}dz}$$ over a circle of radius R centered at the origin enclosing the points z=+a and -a where a>0 (Principal value of ...
2
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1answer
34 views

Integral with residues $\int_0^\infty \tfrac{\sin^2(x)}{1+x^4}dx$

I am trying to calculate $\displaystyle\int_0^\infty \dfrac{\sin^2(x)}{1+x^4}dx$ using method of residues. I have already seen this post, "Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ ...
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1answer
31 views

$\int_{0}^{\infty}{\frac{x^{1/2}\log {x}}{1+x^2}dx}$

integrate in a keyhole contour and show that $$ \int_{0}^{\infty}{\frac{x^{1/2}\log {x}}{1+x^2}dx}=\pi^2/\sqrt(8)$$ and $$ \int_{0}^{\infty}{\frac{x^{1/2}}{1+x^2}dx}=\pi/\sqrt(2)$$ We use the ...
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0answers
12 views

Holomorphic Hermitian metrics

Let $E\to M$ be a complex vector bundle. A hermitian metric $h$ on $E$ is a hermitian inner product on each fiber $E_{p},\, p\in M$. Suppose that $M$ is also a complex manifold and that $E$ is ...
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1answer
32 views

Keyhole Integration $\int_{0}^{\infty}{\frac{x^{k-1}}{(x+1)^2}dx}$

Can you please integrate $$ \int_{0}^{\infty}{\frac{x^{k-1}}{(x+1)^2}dx}$$ using the keyhole integration.I tried to integrate like in $$ \int_{0}^{\infty}{\frac{\log{z}}{(z+1)^2}dx}$$ but I couldn't ...
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2answers
18 views

How to prove this complex binomial power series identity?

I am trying to prove the following: For $k \in \mathbb N$ and complex $z$ such that $|z|<1$: $$ {1 \over (1-z)^{k +1}} = \sum_{n \ge 0} {n+k \choose k} z^n$$ But I can't do it. My first idea was ...
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2answers
45 views

Integral of $\frac{\cos(2x) − \cos(x)}{x^2}$

Will you please solve $$ \int_{0}^{\infty}{\frac{\cos 2x− \cos x}{x^2}dx}$$ using indented contour. I tried like in $$\frac{\sin x}{x}$$ but couldn't figure out.
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1answer
10 views

Evaluating this contour integral

Let $R$ be the rectangle with vertices at $-1$, $1$, $1+2i$, $-1+2i$. Compute $$\int_{\partial R} \frac{(z^2 +i)\sin(z)}{z^2+1}dz$$where the boundary of $R$ is traversed counterclockwise. Here is ...
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3answers
45 views

Prove the identity $\frac{d}{dt} |z(t)|= |z(t)|Re \frac{z'(t)}{z(t)}$

Let $t \mapsto z(t)$ be a complex-valued function of the real variable $t$. Assume further that $z'(t) \not= 0$ for every $t$. Prove the identity $\frac{d}{dt} |z(t)|= |z(t)|Re \frac{z'(t)}{z(t)}$. By ...
3
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3answers
48 views

Using residues to evaluate the integral $\int_{-\pi}^{\pi} \frac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$, $|a|<1$

Calculate the integral for $\left|a\right|<1$ $$\int_{-\pi}^{\pi} \dfrac{\cos(n\theta)}{1-2a\cos(\theta)+a^2}d\theta$$ I'm supposed to evaluate this using method of residues, but the parameter a ...
0
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1answer
10 views

Do these intervals have the same image under the $e^z$ transformation?

Is the image of $D_1=\{z \in C: 0 \lt Re(z) \lt \infty, 0 \lt Im(z) \lt \pi\}$ the same as the image of $D_2=\{z \in C: 0 \lt Re(z) \lt \infty, \alpha \lt Im(z) \lt \beta \}$ as $0 \lt \alpha \lt ...
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1answer
19 views

Determine the order of entire function $G(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^\alpha}$.

Let $$G(z)=\sum_{n=0}^\infty \frac{z^n}{(n!)^\alpha}$$ for $\alpha>0$. Prove that it's an entire function and determine its order. Any suggestions please?
2
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1answer
28 views

Find all holomorphic functions with the following property

Let $D$ be the unit disc. Find all holomorphic functions $f:D\to D$ such that $f(\frac14)=\frac14$, and $f'(\frac14)=\frac7{15}$. I guess that we should use Schwarz lemma. And I guess that the only ...
2
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1answer
24 views

Complex differential operators

Consider the differential operators $\dfrac{\partial}{\partial z}$ and $\dfrac{\partial}{\partial \bar{z} }$ defined by $\frac{\partial}{\partial z} = \frac {1}{2} (\frac{\partial}{\partial x} - ...
4
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2answers
31 views

Prove that if $f$ is holomorphic so that $f'(z)=\alpha f(z)$ then $f(z)=ce^{\alpha z}$

Prove that if $f$ is holomorphic so that $f'(z)=\alpha f(z)$, $\alpha$ being a constant, for every $z \neq 0$ then $f(z)=ce^{\alpha z}$, $c \in \mathbb C$. So what I tried doing is defining ...
0
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1answer
38 views

Problem with a proof of Mittag-Leffler theorem

I've been going through Rudin's Real and Complex Analysis (3rd edition) but I got somehow stuck at the proof of Mittag-Lefler theorem (Theorem 13.10, page 273). The problem is I can't see why Theorem ...
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0answers
28 views

at least one coefficient of the power series is zero [on hold]

Is the following statement is true or false. If $f$ is an analytic function on the disc $\{z\in \mathbb C :|z-1|<1\}$ and $f(z)=f(z^2)$ then $f$ will be constant function. If it is true then why ...
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4answers
30 views

Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions?

Find the image of the unit circle under the transformation $f(z)=\frac{z+1}{2z+1}$. How Do I approach these questions? I tried writing $z$ as $e^{i \phi}$, but I didn't know how to continue from ...
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2answers
18 views

Find all harmonc radial functions.

Find all harmonc functions in C \ {0} wchich are constant on the circles $$ \{ z \in\mathbb{C} : |z| = r \} $$ How to start finding this functions?
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1answer
19 views

The decomposition of a meromorphic function

I want to solve the following problem, taken from the book of T.W Gamelin, Complex Analysis, so it goes like this: Suppose $f(z)$ is a meromorphic on the disk $\{|z|<s \}$ with only a finite ...
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2answers
42 views

Prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded.

Let , $f$ be entire function such that $|f\left(\frac{1}{n}\right)|\le \frac{1}{n^{3/2}}$ for all $n\in \mathbb N$. Then prove that $\{n^2f\left(\frac{1}{n}\right)\}$ is bounded. From the ...
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0answers
36 views

Textbook +reference book in complex analysis

Which book can be used as an introductory textbook in complex analysis? I have some choices (more suggestions are welcomed) Marsden & Hoffman J.B. Conway Ahlfors Palka Lang Stein & ...
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1answer
18 views

Fourier Series of Eisenstein series [on hold]

$$G_{2k}(\tau)= 2\zeta(2k)+2\frac{(2\pi i)^{2k}}{(2k-1)!}\sum_{n\geq 1}\frac{n^{2k-1}q^n}{1-q^n}$$ where $q =e^{2\pi i \tau}$ and $G_k=\sum_{\omega \in L , \omega \neq 0}\frac{1}{\omega^k} $ $L(\tau ...
2
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1answer
31 views

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$

Find all holomorphic functions, $f: \mathbb{C} \rightarrow \mathbb{C}$. so that $f'(0)=1$ and $f(x+iy)=e^{x}f(iy)$ I've been messing with this problem for most of today and haven't managed to get ...
1
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2answers
22 views

Holomorphic curve with unit norm

Is there an open set $U \subset \mathbb{C}$ and a holomorphic function $\gamma: U \to \mathbb{C}^{2}$ such that $\forall z \in U: \parallel \gamma(z) \parallel =1$. if the answer is yes, can the ...
1
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1answer
17 views

Radii problem in a power series

I was studying some basic matters of several complex variables (here $\Omega\subseteq\Bbb C^n$, open): After this, before the proof, the author pointed what follows: So I'm going to tell you ...
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0answers
28 views

‘Integral’ of a Weierstrass $ \wp $-function.

I'm revising for my finals and I've seen a question which asks: Is there a meromorphic function $f: \mathbb{C}/\Lambda \to \mathbb{P}^1$ such that $f' = \wp$? There is a hint which says consider the ...
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3answers
58 views

Complex Number to a power

I asked this question yesterday, but the answers did not actually answer what I wanted to know since I asked the question in the wrong way. I have $e^{i\frac{2014\pi}{12}}$. I know Euler's formula, ...
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0answers
12 views

Classify singularities (poles)

Consider the following function $\frac{1}{\sqrt{h(p-\log(\frac{h}{1-h}))}}$ on where $p$ is real. Are the singularities at $0$ and $\frac{e^{p}}{e^{p}+1}$ removable? essential? Thanks.
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3answers
27 views

Bolzano-Weierstrass theorem (complex case)

I'm trying to prove Bolzano-Weierstrass Theorem to the complex case, i.e., if $(z_n)$ a complex sequence is bounded, then there is a subsequence of $z_n$ which converges. I'm trying to use the real ...
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1answer
23 views

an exercise from “Representation Theorems in Hardy Spaces”, J. Mashreghi [on hold]

$\textbf{Exercise 7.3.2 (page: 165):} \ \text{Let} \ z_0 \in \mathbb{D}, \ \text{and let} \ 0 \leq r \leq 1. \ \text{Show that}$ $$\max_{|z|=r} \bigg| \frac{z_0-z}{1-\overline{z_0}z} \bigg| \leq ...
0
votes
2answers
47 views

Riemann-esque sums (complex analysis)

I have been struggling to prove the following statement: "Let $\gamma:[t_0,t_1] \rightarrow \mathbb{C}$ be a $C^1$ curve. For any $N \in \mathbb{N}$ and $k \in (0,N]$ define $t_N^k := \bigg(1 - ...
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0answers
14 views

Does a relationship exist between the anharmonic absolute elliptical invaliant and the Lattes map?

In a paper in " The Beauty of Fractals" page 153 ,Benoit B. Mandelbrot said his first view of the Mandelbrot set like complex dynamics was in the Samuel Lattes Map which he gives as: ...
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0answers
14 views

What does it mean to be a “family of parametrizations?”

I am reading this book and there is a definition of a curve as follows: So, according to the authors what is a curve? (I am getting confused by "family of all parametrizations" part of the ...
0
votes
2answers
25 views

Understanding Maximum Principle

Understanding Maximum Principle One of the point of that theorem is: If $f$ is analytic on the open connected set $\Omega$ and $|f|$ assumes a local maximum at some point in $\Omega$, then $f$ is ...
1
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2answers
32 views

How to calculate $\int_{C(0,1)}\frac{\sin z}{z^4}dz$

$\displaystyle\int_{C(0,1)}\frac{\sin z}{z^4}\:\mathrm{d}z$, where $C(0,1)$ is the circle around $0$ with radius $1$ $\displaystyle\int_{C(0,1)}\frac{\sin ...
0
votes
0answers
25 views

Contour Integral Example

In the linked lecture notes below https://math.nyu.edu/faculty/childres/lec12.pdf I don't understand the part where the professor writes $$ \lim_{R\to\infty}\lim_{\epsilon \to ...
5
votes
0answers
85 views

A problem about $e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$

Let $\alpha_i\in [0,1),\; i\in \{1,\cdots,N\}$ for some positive integer $N$, such that $$e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$$ and if for any non-empty proper subset ...