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

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
22 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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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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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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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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9 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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2answers
2k views

How to find branch points

I'm solving a set of exercises to understand how to find branch points and branch lines, but I'm having trouble with the more difficult('ish) ones. What I usually do in more simple exercises, is that ...
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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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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
52 views
+50

how to show this function has zeros interlacing and including those of Riemann zeta

Let $\chi (t) = H \left( - \frac{i}{2} (2 t - 1) \right) = \dfrac{4 i \pi \zeta (t) \left( \left( \ddot{\Psi} \left(\frac{t}{2} \right) - \ddot{\Psi} \left( \frac{1}{2} - \frac{t}{2} \right)\right) ...
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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
106 views

How do find the numerical average of $x^x$ from $(-4,-2)$?

I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all complex points and need average to be a real number. To first solve this I found ...
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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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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, ...
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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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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
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 ...
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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
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 ...
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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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2answers
36 views

Good description of orbits of upper half plane under $SL_2 (Z)$

It's known that $SL_2(Z)$ acts on $H=\{z\, |\, Im(z)>0\}$, is there a good description of orbits of $i$ and $w$, other than directly write down $=\{ \frac{ac|z|^2+bc\bar z+adz+bd}{c^2|z|^2+dc\bar z ...
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1answer
22 views

Contour integral and primitive

Given $$f(x,y)=\frac{2}{i(1-y)-x}$$ I have to integrate $f$ over the origin-centered circle of radius 4. I see that $$f(x,y)=-\frac{2x+i2(1-y)}{x^2+(1-y)^2}$$ There is a singularity in $-i$ so I ...
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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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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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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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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 ...
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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
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
52 views

Prove a complex function

Question: Show using the $\epsilon -\delta$ definition that
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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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1answer
46 views

Using the Maximum Modulus Principle to prove that every holomorphic function on a compact Riemann surface is constant

I have read in a number of sources (including here) that a holomorphic function on a compact Riemann surface must be constant. The reason given has always been the Maximum Modulus principle, but ...
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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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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 ...
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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 ...
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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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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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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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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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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
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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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?
3
votes
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 $|1-z|^2$ ...
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 ...
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votes
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 ...