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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38 views

Residue derivation for integral of sin x/x

I am trying to integrate $e^(iz)/z$ over a rectangle contour from $-R$ to $r$ along the real axis and a small semicircle from -r to r and then goes along the real axis to R and then completes the ...
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1answer
19 views

Show that there is $\delta>1$ such that $b$ is analytic on $D(0,\delta)$

For integers $j=1,\dots,n$, let $\alpha_j\in\mathbb{C}$ with $|\alpha_j|<1$ and let $\theta\in\mathbb{R}$, we define $$b(z)=e^{i\theta}\prod_{j=1}^n\frac{z-\alpha_j}{1-\overline{\alpha_j}z}.$$ a) ...
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2answers
68 views

Fractional Linear Transformation: Region between two circles to strip

I'm trying to find Fractional Linear Transformation (if one exists) that maps the region between the circles $\|z+1| = 1\}$ and $\{|z|=2\}$ to the region between the horizontal lines $Im(z) = 1$ and ...
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1answer
29 views

Residue Calculation Question

If I have a function such as $e(2z)/cos(z)$ for example and then I know that there are an infinite number of poles which occur when cos(z)=0. How would I go about calculating the residues for each of ...
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1answer
27 views

show $\sum_{i=1}^Nx_i\bar{y_i}$ is defined

I need to show that $\sum_{i=1}^\infty x_i\bar{y_i}$ is defined if $\sum_{i=1}^\infty x_i $ and $\sum_{i=1}^\infty y_i$ are convergent. this is my solution: since $\sum_{i=1}^Nx_i$ converges then ...
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1answer
37 views

Complex analysis Limit

Find $$\lim_{z \to 1+i} \frac {z^{2} -4z+4+2i}{z^{2}-2z+2}$$ Hi guys, I am now into complex analysis topics, and I have encounter a problem with this limit, What I have tried is to substitite ...
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2answers
45 views

Find all holomorphic functions giving $ u(x,y) = \phi (x^2+y^2) $

Find all holomorphic function , $ f= u+iv $, if $$u(x, y)=\phi(x^2+y^2)$$ As far as I know, for a function two be holomorphic, it has to be harmonic, so $$ \frac{\partial^2 u}{\partial x^2} + ...
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1answer
58 views

Explicit formula for $\zeta$

I can not find anywhere on the internet a proof of this formula : $$\sum_{p\in\mathbb{P}, \; m\geq 1, \; p^m < x} \ln p = x - \sum_\rho \frac{x^\rho}{\rho} - \ln 2\pi - \frac{1}{2}\ln ...
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2answers
236 views

Why does $\int_0^{\infty}\frac{\ln (1+x)}{\ln^2 (x)+\pi^2}\frac{dx}{x^2}$ give the Euler-Mascheroni constant?

I'd like to see the reason why $$\int_{0}^{\infty}\frac{\mathrm{ln}(1+x)}{\mathrm{ln}^2(x)+\pi^2}\frac{dx}{x^2}=\gamma$$ where $\gamma$ is the Euler-Mascheroni constant. I don't have any 'neat ...
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1answer
42 views

Example of conformal map that is not holomorphic?

From an answer to my previous question I understand that if $f: \mathbb C \to \mathbb C$ is holomorphic then it is conformal. I suppose that the other direction of the implication does not hold. ...
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26 views

Taking integral of the complex logarithm using fundamental theorem?

Is it valid to do this? I have $f(z)= z^i$,and $F(z)=\frac{z^{i+1}}{i+1}$ and assuming we're using principle values of $f$ and $F$ would it be correct to say that: $\int_{-1}^{1} f(z) dz = ...
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3answers
84 views

General forms of Harmonic functions

I wish to find the harmonic functions, $u(x,y)$, of the following form $$u(x,y) = \phi(\frac{x}{y})$$ and $$u(x,y) = \phi(\frac{x^2+y^2}{x^2})$$ where $\phi$ is a certain real-valued unknown function. ...
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2answers
25 views

What is gained by moving a complex map into the extended plane?

The Wiki article about Moebius transforms states: A Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the ...
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1answer
46 views

how to show that $f(\mathbb C)$ is dense in $\mathbb C$?

Let $f$ an holomorphic function not bounded. How can I show that $f(\mathbb C)$ is dense in $\mathbb C$ ? I'm sure we have to use Liouville theorem, but I don't see how.
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1answer
40 views

Uniform convergence of a complex power series on a compact set

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ I ...
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2answers
64 views

Laurent Series Expansion computing terms

I need to compute the -5th term to the 5th term of the Laurent expansion of $(\cos(z))^2/\sin(z)$. I know that I can make this into $\csc(z)-\sin(z)$ but I wouldn't know what to do with the $\csc(z)$ ...
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5answers
77 views

Limit with number $e$ and complex number

This is my first question here. I hope that I spend here a lot of fantastic time. How to proof that fact? $$\lim_{n\to\infty} \left(1+\frac{z}{n}\right)^{n}=e^{z}$$ where $z \in \mathbb{C}$ and ...
2
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2answers
47 views

Suppose that $f$ is analytic on a close curve γ. Prove or disprove $\int_\gamma \overline{f(z)}f'(z)dz$ is purely imagine.

Suppose that $f$ is analytic on a close curve γ. Prove or disprove $$\int_\gamma \overline{f(z)}f'(z)dz$$ is purely imagine. I know that $f$ is analytic on a close curve, then $$\int_\gamma f(z) ...
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25 views

Indeterminacy of complex numbers

If $f(z)=\frac{|z|(1+z^2)^4}{(z+1)^2}$ calculate $f'(0)$ I did: $g(z)=|z|\rightarrow g'(z)=\frac{z}{|z|}$ and $h(z)=\frac{(1+z^2)^4}{(z+1)^2}\rightarrow h'(z)=\frac{(z^2+1)^3[6z^2+8z-2]}{(z+1)^3}$ ...
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4answers
201 views

Let $∑_{n=0}^∞c_n z^n $ be a representation for the function $\frac{1}{1-z-z^2 }$. Find the coefficient $c_n$

Let $∑_{n=0}^∞c_n z^n $ be a power series representation for the function $\frac{1}{1-z-z^2 }$. Find the coefficient $c_n$ and radius of convergence of the series. Clearly this is a power series with ...
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1answer
81 views

Is there a novel way to integrate this without using complex numbers?

I've been reading a post on Quora about lesser known techniques of integration and I'm just curious if there's also a novel way to integrate this type of integral without resorting to complex ...
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3answers
89 views

Compute $\sum_{n=0}^\infty \frac {\sin ((2n+1)\phi)}{2n+1}$ for $0<\phi<\pi$

Compute $\sum_{n=0}^\infty \frac {\sin ((2n+1)\phi)}{2n+1}$ for $0<\phi<\pi$ I just want to make sure I did this one correctly. Can I do this $$\sum_{n=0}^\infty \frac {\sin ...
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1answer
54 views

Solve $\sqrt{5-12i}$ by square root definition

I KNOW it can be solved by the trig formula, but I want to solve it by the square root definition, so please don't just post an alternative way to do it. By the square root definition: $$z = 5-12i$$ ...
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1answer
38 views

Complex integration parametric form

Evaluate$\int_{\gamma(0;1)} \frac{\cos z}{z}dz$. Write in parametric form and deduce that$$\int^{2\pi}_0 cos(\cos\theta)\cosh(\sin\theta)d\theta=2\pi$$ By Cauchy's integral formula, ...
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1answer
18 views

Mobius transformation satisfying certain properties

I'm having some trouble showing that a Mobius transformation $F$ maps $0$ to $\infty$ and $\infty$ to $0$ iff $F(z)=dz^{-1}$ for some $d \in \mathbb{C}.$ Mainly with the "only if" part. Do I need to ...
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1answer
46 views

What is a good example of an algorithm that is hard to parallelise?

When I have 10 computers, the factorization of a number doesn't scale along. I am not sure how much faster it would go compared to a single computer, but not 10 times faster like one would expect. ...
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76 views

Compute $\sum_{n=1}^\infty \frac{\cos(n\phi)}{n}$ where $0<\phi<\pi$

Compute $\sum_{n=1}^\infty \frac{\cos(n\phi)}{n}$ where $0<\phi<\pi$ From advance calculus I learn that , I need to find the partial sum and stuff. But from complex analysis class, my professor ...
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3answers
83 views

Radius of convergence of the power series $\sum_{n=1}^{\infty}a_nz^{n^2}$

Find the radius of convergence of the power series $$\sum_{n=1}^{\infty}a_nz^{n^2}$$ where , $a_0=1$ and $a_n=\frac{a_{n-1}}{3^n}$. My Work : We, have, ...
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0answers
119 views

what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$

If $\frac{a}{a+i}+\frac{b}{b+1}+\frac{c}{c+1}=1$ then what is the the value of $\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}$ here I got $a=0$ and $bc=1$, when $ bc\neq 0$ but then I cant ...
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1answer
80 views

Infinite product converges to meromorphic function

How do you show that $\frac{1}{z}\prod_{n=1}^\infty \frac{n}{z+n}(\frac{n+1}{n})^z$ is meromorphic? Any hints would be helpful, I'm having trouble bounding the functions and their logarithms. This is ...
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2answers
83 views

Is the Riemann surface for the square root simply connected?

I am looking for universal covering spaces and I am now wondering if the Riemann surface for the square root $z^{1/2}$ (or even more general for $z^{1/n}$) is simply-connected and therefore a ...
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2answers
52 views

Trouble understanding algebra in proof

Can someone explain to me the step where the integral lower bound turns from n to 1? I was trying to read this proof and I am having trouble understanding that step.
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3answers
45 views

What is the modulus in $\mathbb{C}^3$?

I'm a little confused as to how to assume the form of the magnitude of a vector in $\mathbb{C}^3$, which seems to blur together the concepts of complex and vector modulus. Must the result be strictly ...
2
votes
2answers
61 views

Solution of this non-linear PDE

What is the general solution $h: \mathbb{R}^2 \rightarrow \mathbb{C}$ (or maybe $h: \mathbb{C}^2 \rightarrow \mathbb{C}$ if necessary), $(x, y) \mapsto h(x, y)$ to the non-linear partial differential ...
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1answer
68 views

The function $f(z)=|z|^2$ is only differentiable at the origin

Show that a complex function $f(z)=|z|^2$ is continuous on all complex plan $\mathbb{C}$, but it is only differentiable at the origin. I know that a complex function is continuous at $z_0$ ...
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3answers
93 views

The Laurent series of $(1/(z^2+1)^2$ in the annulus $0<|z-i|<2$

I can't figure it out how to solve this problem: Find the Laurent Series of the function $$f(z)=\frac{1}{(z^2+1)^2}$$ valid in $A=\{z \in \mathbb{C} : 0 < |z-i|<2\}$ I think that it is ...
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2answers
33 views

the order of zero of rational function

let's say at point $z=a$ we have zero of order $n$ to function $f$. My Question is if there is fast way of knowing the order of zero to function $\dfrac{f}{g}$ in point $a$ where $g(a) \neq 0$ ?
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25 views

Asymptotic analysis of $\int_{0}^{\infty} \frac{\sqrt k J^2_{\ell}(k) \sin{(\tau\sqrt k)}}{(k+1/2)^{n+2}} dk$

Question as the title showed, in which $n$ and $\ell$ are positive integers, $\tau$ is real number and $J$ means Bessel functions. How to do the asymptotic analysis when $\tau$ approaches zero? Any ...
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1answer
30 views

Set of Convergence for This Series

What is the set of convergence for this series: $$ \sum_{n=1}^{+\infty} \bigg(\dfrac{(2n-1)!!}{(2n)!!}\bigg)^3\bigg(\dfrac{3z-1}{2}\bigg)^n $$ My initial thought is to use, $ \dfrac{1}{R} = ...
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1answer
87 views

Calculating Residues and singularities

We haven't done many full examples with residues so just wondered how you would answer the following questions for example: Classify the type of singularity for f and determine the residue of f at ...
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2answers
27 views

Set of Convergence for the following Series

What is the set of convergence for this series: $ \sum_{n=1}^{+\infty} \dfrac{3^{\sqrt{n}}(2+i-3z)^n}{\sqrt{n^2+1}} $ ? My initial thought was to use, $ \dfrac{1}{R} = \lim(|a_n|)^{1/n}$, but this ...
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1answer
27 views

How to evaluate $\frac{1}{2\pi i}\int_C \frac{f(z) dz}{(z-z_1)^{m_1}(z-z_2)^{m_2}}$ and the other related contour?

Respected All. I was studying residue theory where I came accross the following problem "If $f$ be analytic in the simply connected domain $D$ and $z_1, z_2$ are two distinct complex point lying in ...
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37 views

Zeroes of Jacobi Theta Functions

Based on wolfram alpha: $$\sum_{i=0}^{\infty}[x^{i^2}] = \frac{1}{2}(v_3(0,x) +1) $$ Whereas $v_3$ is the third Jacobi Theta function. See: http://bit.ly/1FIyUTq I am curious for what values in ...
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23 views

How to calculate the product of $\sin\frac{\pi}{n}$ to $\sin\frac{(n-1)\pi}{n}$ [duplicate]

How to calculate $\sin\frac{\pi}{n}\sin\frac{2\pi}{n}\cdots\sin\frac{(n-1)\pi}{n}$?
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1answer
52 views

Deriving the Poisson Integral Formula from the Cauchy Integral Formula

If $f$ is analytic inside and on the unit circle $\gamma$, show that for $0<|z|<1$, $$2\pi if(z)=\int_\gamma \frac{f(w)}{w-z}dw-\int_\gamma \frac{f(w)}{w-1/\bar{z}}dw$$ and then derive the ...
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0answers
38 views

Function not analytic anywhere

It can be shown rather easily that the function $f(z)= x^2 +iy^3$ is not analytic anywhere ...But then why are the Cauchy - Riemann equations satisfied at $x=0$ , $y=0$ ???
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46 views

distribution of eigenvectors of a random matrix

Let $\mathbf{A}\in\mathbb{C}^{n\times n}$ be a random hermitian matrix. Assume that the eigenvalues of this matrix have continuous probability distribution. 1.Can we say that the eigenvectors ...
2
votes
1answer
63 views

Is the derivative formula itself complex differentiable?

Let $f$ be complex differentiable on an open set $U\subset\mathbb{C}$ (and therefore holomorphic on said set, according to my teacher, since in my class we consider this implies the derivative is ...
2
votes
1answer
55 views

Winding number (demonstration)

How could I explain mathematically, that the winding number of a closed curve $\gamma$ around $a$ ($a \notin \gamma$) gives always an integer value. $$ W(\gamma,a)=\frac{1}{2\pi i} \int_{\gamma} ...
1
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2answers
102 views

Mobius transformation from region between two intersecting circles to an annulus

I'm trying to find a Mobius transformation from from the region between the circles $|z|=1$ and $|z+1| = \frac {4}{\sqrt(3)}$ to an annulus. I've tried to find three points in the original region that ...