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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32 views

Rouche's theorem equivalent for reals

One of the best things I liked about my Complex Analysis course were all the great tools we had to find roots, integrals etc. some tools we couldn't have used in $ \mathbb{R}$. I know we can use the ...
2
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2answers
54 views

Help finding the residue of $1/(z^8+1)$

Help finding the residue of $1/(z^8+1)$ I'm integrating over $\{ Re^{it} | 0 \leq t \leq \pi \}$, and I found 4 simple poles at $z_0=e^{in\pi/8}$ where $n = 0,...,3$ and I'm trying to calculate ...
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1answer
32 views

Calculating complex derivative from definition

I have to check differentiability of $f(z)=\mathbb{Re}^3z+\mathbb{Im}^3z$ at $0$. $$f'(0)=\lim\limits_{(x,y)\to (0,0)}\frac{f(x+iy)-f(0)}{x+iy}$$ $$0\le|\frac{x^3+y^3}{x+iy}|\le ...
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1answer
29 views

equation draws a circle in C

It be $a\in\Bbb C , b ∈\Bbb R$ and $|a|^2 > b$. Also, $a'$ is the conjugation of $a: a' = x - iy$ when $a = x + iy$ (and equally for $z$). It needs to be shown, that the solutions of the equation: ...
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1answer
38 views

Question about complex exponentation

We define: $z^w:=e^{wLogz}$ but in general for complex numbers $wLogz \ne Log z^w$. (Even if we take principial branch). So what is the reasoning for it?
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29 views

“The limit of a sequence is insensitive to finite changes in the sequence” - help me understand this sentence!

The context is p59 of "A primer on Hilbert Space Theory, by Alabiso and Weiss, where the example is given of a quotient $c_{00}/c_0$ which is "the quotient of the space $c_0$, the space of all ...
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2answers
41 views

Find the slope at $\theta = \pi/3$

We are given $r = 2\cos(\theta)$ Find the slope at $\theta = \pi/3$ This is a polar curve. Why would you find the slope at $dy/dx$ rather than $dr/d\theta$?
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1answer
15 views

Laurent series expansion - inequalities of series coeffiients

Let $f:U\rightarrow \mathbb{C}$ be analytic. Suppose $\overline{A}_{r,R}(z_0)\subset U$ where $r<R$ and $r,R \in (0,\infty)$. Also, $|f(z)|\leq M$ in $\overline{A}_{r,R}(z_0)$. I need to prove the ...
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1answer
66 views

Laurent series, integral over the annulus, radii

We are given $$f = \sum_{n= - \infty} ^{\infty} a_n (z-z_0)^n \in \mathcal{O} ( \text{ann} (z_0, r, R)), \ \ 0<r<R< \infty. $$ Prove that $$\frac{1}{\pi} \int _{ann (z_0, r, R)} |f(z)|^2 d ...
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3answers
22 views

Limit About Complex Variable

I am trying to see if I have right understanding of the limit. when the book mentions $$\lim_{z\to z_0} f(z) = L$$ this just means that $f(z_0) = L$ is this correct?
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1answer
36 views

Do real-analytic functions always extend uniquely to complex-analytic functions on $\mathbb{C}$?

A function $f(x)$ is an real function and analytic in an open interval of $x$-axis or the whole $x$-axis. Is there only unique way to analytically extend it to the whole complex plane? I know ...
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1answer
24 views

Does there exist a non-constant complex function $f(z)$ which is analytic in the whole $z$-plane and infinity point?

Does there exist a non-constant complex function $f(z)$ which is analytic in the whole $z$-plane and infinity point? Or must a function which is analytic in the infinity point be singuler in some ...
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0answers
17 views

Logarithm in Complex Plane and Open Set

I am still kind of confused about logarithms in complex plane. If you cut off the one of the line such as positive x-axis. I understand every set excluding positive x-axis will be holomorphic and ...
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2answers
35 views

Need help with problems in complex analysis about analyticity

Show that $\nexists f\in H(B_1(0))\cap C(\bar{B}_1(0))$ such that $f(z)=\bar{z}$ on $\delta B_1(0)$ i.e., there exists no $f(z)$ analytic on open unit circle and continuous on closure of unit circle ...
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1answer
34 views

Proving $f(z)$ entire function in complex analysis

If $f\in C(\Bbb C)\cap H(\Bbb C\backslash \delta B_1(0))$ then $f\in H(\Bbb C)$ [C means continuous, $\Bbb C$ means complex plane, H means analytic and $\delta$ means boundary] I don't even know ...
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1answer
45 views

Problem in complex analysis

Let $G$ be a bounded region and $f\in H(G)\cap C(\bar{G})$ [$H$ means analyticity and $C$ means continuity]. If $|f(z)|$ is constant on the boundary of $G$. prove either $f(z)$ has a zero in $G$ or ...
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2answers
70 views

complex analysis problems related to nalyticity… [on hold]

I don't even know how to start for the following problem. It would be highlt appreciated if you could help me. Let $z_n=\frac{1}{n},\forall n\in\Bbb N$. If possible find $f\in H(\bar{B}_1(0))$ [H ...
2
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1answer
57 views

let $f$ be analytic and bounded above, can I prove f is constant?

I've read up on Lioville's theorem and I was wondering if this could also be proved using the theorem: let $f$ be analytic on $\mathbb{C}$ and let $K>0$ be s.t. $|f| \geq K$, could I prove using ...
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1answer
43 views

Path integral in the complex plane

Evaluate $\int_Tz\,\mathrm dz$ and $\int_T\overline z\,\mathrm dz$ where $T$ is the triangle with vertices $0,1,-i$ oriented clockwise. I am trying to solve this question, but I'm unsure how to ...
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1answer
21 views

Statement of maximum modulus principle and question

Question 1 My book (Complex Variables by Churchill) states If a function $f$ is analytic and not constant in a given domain $D$, the $|f(z)|$ has no maximum value in $D$. that is, there is no ...
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0answers
25 views

Conformal Map for Circle to Circle

I am trying to find a conformal map that maps a circle in the $\zeta$ plane to a circle in the $z$ plane. As far as I know, a Mobius transformation is appropriate for this. These are the conditions ...
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3answers
20 views

Taylor Series of $\frac{\log(z+1)}{z+1}, z_{0}=0 $

Finding the Taylor Series of $\frac{\log(z+1)}{z+1}, z_{0}=0 $ Where Log is the complex logarithm. At first, I tried to find the series for $\log(z+1)$ and $\frac{1}{z+1}$ and multiply them. ...
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38 views

Construction of an explicit series of meromorphic functions.

I have to construct an explicit series of meromorphic functions that converges locally uniformly on the unit disk, and such that it has poles of first order at the points $a_k:=\frac{k-1}{k}$ with ...
2
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1answer
35 views

When I can use Cauchy's Int. Formulae

Let $C$ be the circle $|z| = R$ traversed anticlockwise. Let z be any point in $\mathbb{C}$. I'm asked to calculate: $$\frac{1}{2\pi i} \int_C \frac{e^w}{w^3}(w^2+wz+z^2) \ dw$$ I'm wondering here ...
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2answers
64 views

If $f(z)g(z)=0$ then $f(z)=0$ or $g(z)=0$ [duplicate]

Let $D$ be a domain and let $f,g$ be analytic in $D$. I need to prove that if $f(z)g(z)=0$ for all $z\in D$, then $f(z)=0$ for all $z\in D$ or $g(z)=0$ for all $z\in D$.This is my answer. Assume ...
2
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1answer
33 views

When does this complex series converge?

Let $$f(z)=\sum_{n=0}^{\infty}\frac{z^{n [\mathbb{Re} z]}}{n}$$ For which $z\in\mathbb{C}\setminus\{0\}$ does this series converge? I have trouble with this example. When I use the ratio test, I ...
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2answers
46 views

Need help with complex analysis

I am struggling with following problem: Let f be an entire function, If $Im(f)>|z|$ for all $|z|>2$, show that f is constant. I think I have to use Liouville's theorem to do this problem. That ...
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0answers
12 views

An inequality in Zygmund space

How to derive (3) from (2)? Thanks for help.
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0answers
22 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
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3answers
63 views

Video Lessons in Complex Analysis

Does anybody have some link for good video lessons of a complete course in Complex Analysis? Grateful.
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1answer
27 views

A question about the properties of the pseudospectrum

Assume that $A\in \mathbb{C}^{n\times n}$. The $\epsilon-$pseudospectrum of $A$ is defined by $$\sigma_{\epsilon}(A)=\{z\in C \quad | \quad \Arrowvert (zI-A)^{-1} \Arrowvert>\frac{1}{\epsilon}\}.$$ ...
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2answers
50 views

Which entire functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$?

Which holomorhpic functions satisfy $\,\lvert\,f(z)\rvert \leq \lvert z\rvert^k$ on $\mathbb{C}$? So I've shown that $\,\lvert\, f(z)\rvert \leq |z|^k \implies f(z)$ is a polynomial of degree at most ...
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0answers
12 views

Determine all of Möbius transformations $S$ such that $S(D(0,1))=D(0,1)$. [duplicate]

Determine all of Möbius transformations $S$ such that $S(D(0,1))=D(0,1)$. I thank you for your help.
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0answers
33 views

Quadratic formula and complex numbers

Let $az^2+bz+c=0$ be a complex quadratic equation. We know that it has $2$ roots: $z_1=\frac{-b-\sqrt{b^2-4ac}}{2a}$ and $z_2=\frac{-b+\sqrt{b^2-4ac}}{2a}$ If $b^2-4ac=1+i$ for example we have to ...
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2answers
71 views

Prove that if $\,\lvert\, f(\,f(z))\rvert>r,\,$ then $f$ is constant

Let $r>0$. Prove that if $f$ is holomorphic on a whole complex plane and $|f(f(z))|>r$ for all $z\in\mathbb{C}$, then $f$ is constant. Can sb point me in the right direction?
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1answer
34 views

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$ for $n=2,3,\dots$ Can somebody point me in the right direction? Only thing I know is that $f(0)=0$ if such ...
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1answer
36 views

definition of derivative for complex analysis

How can I use the definition of derivative to find the derivative of $\dfrac{\bar{z}^2}{z}$. My attempt, $\dfrac{\dfrac{\overline{z+\Delta z}^2}{z+\Delta z}-\dfrac{\bar{z}}{z}^2}{\Delta z}= ...
2
votes
2answers
29 views

Finding the Laurent series of $\frac{1}{z^{2}(1-z)}, 1<|z|<\infty $

Finding the Laurent Series of $\frac{1}{z^{2}(1-z)}, 1<|z|<\infty $ I tried to divide both numerator and denominator by $z^2$, so that there is a term of $\frac{1}{(1-z)}$. I am tempted to ...
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1answer
28 views

Laurent Series of $\frac{e^{z}}{(z+1)^{2}}, 0<|z+1|<\infty $

$\frac{e^{z}}{(z+1)^{2}}, 0<|z+1|<\infty $ I am utterly unable to solve this problem. I have tried to write it as $e^(z-2\ln(z+1))$, but the resulting series is completely hideous. I have ...
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1answer
16 views

Show that $H(\mathbb{C})$ is a Frechet space.

Let $H(\mathbb{C})=\{f: \mathbb{C}\rightarrow \mathbb{C}; f \text{ holomorphic}\}$. For each $n$ let the seminorm $p_n$ be $p_n(f)=\sup_{|z|\leq n}|f(z)|$, and let $d(f,g)=\sum_{n=1}^\infty ...
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2answers
38 views

Bounding a real integral involving complex constant

Is this integral finite $$|\int_{-\infty}^{\infty} e^{-i\pi x^2}\ dx|$$ can we use the fact that $e^{-\pi x^2}$ have compact support to estimate the above integral?
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0answers
50 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
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1answer
70 views

$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq ...
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2answers
48 views

$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that ...
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votes
1answer
56 views

Let $f(z)=z+\frac 1 z$ for $z\in \mathbb C$ with $z\neq 0$ [on hold]

Let $f(z)=z+\frac 1 z$ for $z\in \mathbb C$ with $z\neq 0$. Which of the following are true? $f$ is analytic function on $\mathbb C\setminus \{0\}$. $f$ is a conformal map on $\mathbb C\setminus ...
3
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1answer
73 views

Riemann-Roch Theorem

Could somebody give a simple plain English explanation as to what the Riemann-Roch theorem is about to somebody who knows only standard one-variable complex analysis. Thanks.
0
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1answer
53 views

Minimal surface and Weierstraß parametrization

If I have $f(z) = 1$ and $g(z) = \frac{1}{z}$ and I am looking for a minimal surface on $\mathbb{C} \backslash \{0\}$ using the Weierstraß-Enneper representation of minimal surfaces. Now I was ...
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2answers
55 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
0
votes
0answers
5 views

Time series of Complex numbers and Lyapunav exponent calculation

I have a time series of length $n$ of complex numbers that is $\{(x_i,y_i)\}$. $i=1,2,\dots,n$ where $(x_i, y_i)$ is a complex number. I would like to calculate the Lyapunav exponent of the series. ...
2
votes
2answers
60 views

Using Residue theorem

I read Book by Egorychev in Russian. I don't understand the following identity $$S_m=\frac{1}{2\pi i m}\int_{|w|=\frac12}(2+w)(1+w)^m(1+w+w^2)^{-1}w^{-m-1}dw=-\frac1m ...