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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20 views

How many entire functions are there that are equal to $\frac{1}{z}$ for $|z| > 1$?

Let $f:\{z\in\mathbb{C}:|z|>1\}\rightarrow\mathbb{C},f(z)=\frac{1}{z}.$ Now my question is how many entire functions $g$ are there such that $f=g$ for $|z|>1$?According to me there is no such ...
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1answer
25 views

How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$.

How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$. Definition: Consider a partition $P,a=t_0\lt t_1\lt \ldots\lt t_n=b\; ...
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3answers
27 views

Integration of a real valued function on complex plane

Suppose $f: \mathbb{C}\rightarrow \mathbb{R}$ $f$ is continuous, bounded, $f(z)\geq 0$. Can we claim that the following integration $$\int_{C_R}f(z)dz$$ is equal to zero? ($C_R$ is a circle ...
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1answer
21 views

Jordan curve of infinite length

I was thinking about Jordan curve with infinite length and Koch snowflake seems to be a valid answer intutively. Can anyone give mathematical proof for this?
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1answer
14 views

Prove $h=$Log $z$ in a domain

I'm trying to show that if $h$ is analytic in $D=\{z\in\mathbb{C}: |z-1|<1\},$ $h'(z)=z^{-1},$ and $h(1)=0,$ then $h$ is Log $z$ in $D$. I know Log' $z=z^{-1}$ and Log $1=0.$ However, I don't know ...
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1answer
37 views

Complex Analysis Lectures

I am looking for a series of video lectures on the subject of complex analysis which follow Conway's text Functions of Complex Variable I. Any recommendations?
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10 views

Limit argument for complex squareroot

Let $z\in \mathbb{H}\backslash(0,i]$, where $\mathbb{H}$ is the upper half plane. I want to show that $z(\sqrt{z^2+1}-z)\rightarrow \frac{1}{2}$ for $z\rightarrow \infty$, where $w\mapsto \sqrt{z}$ ...
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0answers
10 views

Optimizing functions with a complex domain and a real codomain

In general I want to understand if it makes sense to optimize a function of the following form $f: \mathbb{C} → \mathbb{R}$ for my specific problem $f(z) = | z | ^{2} $ (wich I is not analytic since ...
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0answers
14 views

technique of proving continuity complex function

If I want to prove a function $f(t)$ is continuous at $ t_o$. By def of continuity, I need to show that for $\epsilon >0 $, there exist $ \delta>0$ s.t $ |t-t_o| < \delta$ implies $ ...
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0answers
25 views

Describe set of $z^2$ as z moves over 2nd quadrant and show it is open and connected

Problem: Describe the set of points $z^2$ as $z$ varies over the second quadrant: {z = x + iy; x < 0 and y > 0}. Show this is an open connected set. (Hint: use the polar representation of z.) The ...
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0answers
19 views

Zeros of this function?

Let $$f(z)=\gamma + z^{\beta_2-\beta_1}$$ where $\gamma\in \mathbb{R}$, $\beta_1\in \mathbb{Z}$, $\beta_2 \in \mathbb{Z}$ and $\beta_2 > \beta_1$. The variable $z$ takes complex values. Is there a ...
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4answers
46 views

Laurent series of $\dfrac{1}{\sin(\frac{1}{z})}$ [on hold]

What is the Laurent series for the function $f(z)=\dfrac{1}{\sin(\frac{1}{z})}$ at the point $z=\dfrac{1}{\pi}$ and $z=0$?
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0answers
29 views

Problem with $\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$ (by residues) [duplicate]

I, I am trying solve the following integral $$\int_{0}^{\infty} \frac{\log^2(x)}{1+x^2}$$ Teachers teached me that I can solve the integral $$\int_{0}^{\infty} ...
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0answers
15 views

Find the radius of convergence [on hold]

Find the radius of convergence of $\sum\limits_{n=0}^{\infty} \exp(inθ)(θ^n)z^{2n}$ with n running from 0 to infinity, non-zero θ. I would usually apply the ratio test, but with θ in the equation too ...
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0answers
23 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
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0answers
28 views

Homotopic, winding number and continuity, Conway text

I have a question about this. Here is the def of homotopy used by Conway, in case you guys need it. Let $G$ be an open set in $\mathbb{C}$ and let $\gamma$ be a closed smooth rectifiable curve in ...
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2answers
18 views

Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist.

Show that $\lim_{z \to 0} \frac{\Re(z)}{z}$ doesn't exist. Let $z=r(\cos(\theta)+i \sin(\theta))$. So $\frac{\Re(z)}{z} =\cos ^2(\theta) - i \cos(\theta)\sin(\theta) $, and $$\lim_{z \to 0} ...
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0answers
17 views

If there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$

Show that if there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$ I am unable to proceed any further in this and any help in this regard ...
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3answers
51 views

Can a complex argument into this function ever yield a real result?

I have a function defined as: $f(z)=\frac{\Gamma{(z)}+1}{z}$ Are there any $z ∈ C$ (with nonzero imaginary part) such that $f(z)∈R$? I tried substituting in $z=a+bi$ with $b≠0$ into the product ...
2
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4answers
41 views

Analyticity of $\overline {f(\bar z)}$ given $f(z)$ is analytic [duplicate]

Suppose $f$ is an analytic function on a domain $D$. Then I need to show that $\overline {f(\bar z)}$ is also analytic. Here is what I did - Suppose $f(z) = u(x,y) + iv(x,y)$ where $u$ and $v$ are ...
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1answer
33 views

Complex integration on upper-half plane

In order to prove the normalisation property of a Lorentzian function, $L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$ we take a closed contour on the ...
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1answer
36 views

Determine for what values $z \in \mathbb{C}$, $\sum_{n = 1}^{\infty} \frac{z^n}{n^2}$ is convergent.

I am not sure where to start on this one. I know that $z^n$ can be written as $\sum_{n=0}^{\infty} \frac{1}{1-z}$. But I do not know how to proceed.
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0answers
15 views

Branch point at infinity?

I have to find the branch points of $f(z)=\left( z(z+1)\right )^{1/3}$. It is clear that $0$ and $-1$ are branch points, but I am not sure about infinity. Making the substituition $x=\frac{1}{z}$ and ...
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2answers
43 views

Does there exist an analytic function $f$ on $D(0,1)$ such that $f(z_n)=0$ for even $n$ and $f(z_n)=1$ for odd $n$?

Given that $(z_n)$ is a sequence of distinct points in $D(0,1)=\{z \in \Bbb C : |z| \lt 1\}$ with $\lim_{n \to \infty} z_n=0$, Can we find an analytic function $f$ such that $f(z_n)= \begin{cases} 0, ...
0
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2answers
41 views

Is $f( x + iy) = e^{-x} e^{-iy}$ complex - differentiable?

I started by letting $u(x,y) = e^{-x}$ and $v(x,y) = e^{-iy}$ . I then tried to use the cauchy reiman equations : $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial ...
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2answers
33 views

Let $\gamma=\{z\in \Bbb C: \lvert z \rvert=2\}$ in anti-clockwise orientation. Then $I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz$=?

$$I=\frac {1}{2\pi i} \int_{\gamma} z^7 \cos \frac 1{z^2} dz=?$$ The function $\cos \frac 1{z^2}$ is neither analytic at $z=0$ and nor it has a pole at $z=0$. By Cauchy Integral Formula can I get ...
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1answer
30 views

Show that $g(z)=\frac{1}{n}\sum_{k=0}^{n-1} f \left(\xi^{k}\sqrt[n]{z}\right)$ is an entire function.

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f(\xi z)=f(z)$ for all $z\in \mathbb{C}$ and consider the ...
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0answers
23 views

Integral of action (quantum field theory, prescription)

I am struggling to show: $$\int_{w=0}^{w} \int_{r=2M}^{2(M-w)} \frac{-drdw}{1-\sqrt{\frac{2(M-w)}{r}-\frac{Q^2}{r^2}}}=2\pi[{2w(M-\frac{w}{2})-(M-w)\sqrt{(M-w)^2-Q^2)}+M\sqrt{M^2-Q^2}}]\\$$ with ...
2
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1answer
33 views

Complex derivative numerically using real $h$ and imaginary $h i$?

I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like $$ \frac ...
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0answers
19 views

Proof involving complex limits

Prove that $\lim_{n \to \infty } \left | z_{n} - z \right | = 0$ if and only if $\lim_{n \to \infty } Re(z_{n}) = Re(z)$ and $\lim_{n \to \infty } Im(z_{n}) = Im(z)$. I understand the epsilon delta ...
2
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1answer
38 views

Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that…

In reviewing complex analysis, I stumbled upon the following problem: Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when ...
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0answers
39 views

How to find branch points for complex functions?

I'm looking for a standard way I can approach problems where I am tasked to find the branch points and branch cuts of a complex function. For instance, $$ f(z) = e^{(z^2+1)^{1/2}}$$ or $$ f(z) = ...
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votes
1answer
47 views

There is an entire functión $g$ such that $f(z)=g\left(z^{n}\right)$.

Let $f$ be an entire function and $\xi=e^{\frac{2\pi i}{n}}$ for some $n\in \mathbb{N}$. Suppose that $f\left(\xi z\right)=f(z)$ for all $z\in \mathbb{C}$. Show that there is a entire function $g$ ...
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3answers
23 views

Use complex numbers to deduce triple angle formulas [on hold]

How to prove $\cos{3\theta}=\cos^3{\theta}-3\cos{\theta}\sin^2{\theta}$?
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2answers
35 views

Disc of convergence of a power series

Find the disc of convergence: $$\sum_{n=3}^\infty \left(1-\frac{1}{n^2}\right)^{-n^3}z^n$$ I have been manipulating the power series and I am pretty sure it has something to do with $e$ but I cannot ...
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2answers
24 views

Show if one series converges absolutely then so too does the other.

Task at hand: Let $a_n$ and $b_n$ be nonzero complex numbers for $n=1,2,3...$ . Suppose $\lim_{n\to \infty} \left|\frac{a_n}{b_n}\right|=l$ exists, and $l\neq0,\infty.$ show that if one of the series ...
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1answer
26 views

Show that entire function $f$ is a polynomial of degree at most $n$

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ ...
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2answers
28 views

Show that the function $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$.

Show that $\phi (x,y)=\arctan{\frac{2x}{x^2+y^2-1}}$ is harmonic by considering $w(z)=\frac{i+z}{i-z}$. I know that if $\phi$ is harmonic then it satisfies Laplace's equation but I don't see how ...
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2answers
28 views

how could calculate $ \int_{C} \frac{1}{\sin(z)} \, dz $ when $C=C(0,1)$

i am trying calculate $$ \int_{C} \frac{1}{\sin(z)} \, dz $$ when $C=C(0,1)$ by complex methods, its said, by residues, some one could help me?
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1answer
16 views

Equality of analytic functions equal on a diverging sequence of complex

I ask this question as a subsequent of following one. Suppose that $f$ and $g$ are two analytic functions defined on $\mathbb C$ and that $(a_n)_{n \in \mathbb N}$ is a sequence of complex numbers ...
3
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2answers
52 views

Proving limit of $|1-z|^2$ as $z \to i$ is 2

First off, apologies for my formatting. This is my first post and I'm still unfamiliar with MathJax and Latex, so I'm doing the best that I can. So I'm trying to prove that the limit of $|1-z|^2$ ...
1
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1answer
44 views

Questions 4 and 5 from section 4.3 of Conway's complex analysis book

I'm reading the Conway's complex analysis book and I'm trying to solve theses exercises on page 80: 4.Prove that $e^{z+a}=e^ze^a$ 5.Prove that $\cos(a+b)=\cos a\cos b-\sin a\sin b$ The ...
3
votes
1answer
39 views

Calculate $\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$ for $n\in \mathbb{Z}$

Calculate $$\int_{\left|z-1\right|=2}z^{n}\sin\left(z\right)dz$$ for $n\in \mathbb{Z}$ My attempt: According to the following result which was presented at my course as Cauchy's integral formula for ...
2
votes
2answers
27 views

How could I calculate $\int_{C} ze^{\frac{1}{z-1}}$ when $C=C(1,\frac{1}{2})$

I have to solve if $C=C(1,\frac{1}{2})$ $$I=\int_{C} ze^{\frac{1}{z-1}}$$ I know that $I=2\pi i \operatorname{Res}(f(z), 1)$, but I do not know how could I calculate that residue. What I did: ...
0
votes
2answers
28 views

If $x\in \mathbb{R}$ then show that $\{z\in \mathbb{C}: \Im(z) < x\} =A$ is open.

THE RED LINE IS $ \Im(z) = x$ Now, my proof is as follows, Let $z' \in A$, then take $\epsilon = x - \Im(z')>0$ Now let $w \in D_{\epsilon}(z')$ and suppose $w \notin A$ then $$\epsilon > ...
0
votes
1answer
35 views

Supremum of $\cot(\pi z)$ where $z$ is on circle with radius $n+1/2$

I try to estimate the supremum of $|\cot(\pi z)|$ and where $z=(n+1/2) e^{i t}$, $n\in\mathbb N$ and $t\in[0,2\pi)$. I should be a constant. So far I did by wiriting it in exponential form and ...
1
vote
1answer
26 views

Commuting $\operatorname{Re}$ with integral

Is the following always true? $$ f:\mathbb{C}\to\mathbb{C},\ \operatorname{Re}\left(\int f(z)d z\right) = \int\operatorname{Re}(z)dz $$ $$ \frac{d\operatorname{Re}(f)}{dz} = ...
0
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0answers
22 views

Lang's proof of the Weierstrass preparation theorem

Relevant Google Books link. I'm having problems with the final step in the proof of Theorem 9.1. It's not clear to me why the function $I + \tau \circ \frac{\alpha(f)}{\tau(f)}$ should be ...
1
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1answer
31 views

Meaning of $\partial f /\partial x$

I have an exercise in complex analysis that begins, If $U\subset \mathbb C$ is an open set and $f:U\to \mathbb C$ is real differentiable.... Later on, it allows me to assume $f$ is holomorphic. ...
0
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0answers
16 views

Runge's Theorem Application

Below is a question out of Gamelin's Complex Analysis which I cannot quite figure out. Any tips would help appreciated! "Let $(z_j)$ be a sequence of distinct points in a domain $D$ that accumulates ...