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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8 views

The image of a specific Mobius transformation

Let $f:D\rightarrow \mathbb{C} :f(z)=\frac{z}{z-1}$ and $D= \{ z:|z|=1\}$ ,what is the image of $f$, $f(D)$? Can one elaborate on some general methods of dealing with these kind of questions?
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1answer
14 views

Meromorphic functions on the unit disk

is there any characterisation of all the holomorphic or meromorphic functions from the open unit disk to itself? As an example of what I mean by characterisation, the holomorphic functions on the ...
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0answers
30 views

Am I using the Residue Theorem Correctly?

I am trying to evaluate $$\int _{C_a}\frac{z^2+e^z}{z^2(z-2)}dz=*$$ where $a>0$ and $C_a$ is a circle of radius $a$ centered at the origin. If $a<2$, I apply the Residue Theorem to obtain ...
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1answer
22 views

laurent series expansion about $z=0$

using the Laurent expansion i got the answer to be $$-(z+1)\sum_{n=0}^\infty \frac{z^{n-1}}{2^{n+1}}$$ however, I've got a feeling I've made a mistake somewhere?
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1answer
18 views

Finding the order and computing the residue of a pole

Find the poles, indicate their order and compute their residues for the following functions: $$g(z)=\frac{e^z}{\sin z}$$ I have a singularity at $z=0$ where the residue would be $1$ ... however, ...
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1answer
15 views

Characteristic Function and Density Function

Consider a random variable $X$ with density function $f(x)$, moment generating function $M(t):= \int e^{tx}f(x) dx$ (existing in an interval containing $0$), cumulant generating function $K(t):=\log ...
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1answer
19 views

How transformation of co-ordinates system relates to its vectors?

Consider a positive definite matrix. Can we consider that it has a underlying co-ordiante system? If we transform that co-ordinate system how the the vectors are transformed? Is this question even ...
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18 views

Winding number is locally constant

Let $\gamma$ be a closed path in the plane $\mathbb{C}$ and let $a\in \mathbb{C}$ which does not belong to the image of $\gamma$. The winding number (or index) is defined as $$I(\gamma, ...
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38 views

Radius of convergence and sum of alternating series $1 - z + z^2 - z^3 + \ldots $

I have a (complex) function represented by the power series \begin{equation*} L(z) = z -\frac{z^2}{2} + \frac{z^3}{3} - \frac{z^4}{4} \ldots \end{equation*} which I have tried to represent (perhaps ...
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1answer
16 views

Meaning of co-ordinate system of Covariance matrix

Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the ...
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0answers
20 views

Proving a version of maximum modulus principle elementarly.

There is this version of maximum modulus principle: If $P$ is a non-constant polynomial, then $|P|$ doesn't have a local maximum. I know that if $P$ is non-constant, then $|P(z)| ...
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1answer
48 views

If $ g \circ f$ is real analytic and $g$ is a real analytic immersion, then $f$ is real analytic

Let $M$ $N$ $P$ be complex manifolds, and let $$f: M\rightarrow N, g: N\rightarrow P$$ be $C^\infty$ maps with $g$ and $g\circ f$ holomorphic, and with $dg$ never degenerate. It's easy, then, to see ...
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222 views

What's wrong in this equation? (Regarding Euler's eqn)

I got an idea, but that doesn't match with Euler's theory.. So What's wrong?! $$e^{jx} = (e^{j 2\pi})^{x/2\pi} = 1^{x/2\pi} = 1$$
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0answers
27 views

Help for solving limi of the Complex Fourier Series

I need help for this exercise. Let: $ f:\left[ -T /2, T/2 \right]\rightarrow \mathbb{R}. $ I need show that $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
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2answers
33 views

How to solve this complex logarithm equation?

define $Log z := ln|z| + i Argz$ and solve the equation $Log(z^2-1)=i \pi/2$, for all possible value I've try that let $w=z^2-1$and $Log\ w = i\pi/2$, then $|w|=1$and$Arg\ w=\pi/2$ ...
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1answer
45 views

Prove that $f$ has a simple pole at $z=0$

Let, $f:\{z\in \mathbb C:0<|z|<1\}\to \mathbb C$ be analytic such that $n\le |f(1/n)|\le n^{3/2}$ for $n=2,3,...$. Assume that $z^2f(z)$ is bounded in $|z|<1$. Show that $f$ has a pole of ...
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22 views

Computing an integral using residues

I am trying to find an integral $ \int_{-\infty}^{+\infty} \frac{ e^{-(x^2 + 1)}}{(x^2 + 1)^2} dx $. I went about applying contour integral over a semicircle with diameter along $ x = +\infty$ to ...
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1answer
25 views

How is the boundary in product spaces defined?

The general question: how is the boundary defined in product spaces? Given two topological spaces $X,Y$, I'd say that $\partial(X\times Y)=\partial X\times\partial Y$. But looking at what follows it ...
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1answer
20 views

How to use Cauchy's integral formula with more than one pole?

$\int\limits_{\gamma} \frac{z^2}{z(z-2)}$ $\gamma(\theta) = 3e^{i\theta}$, $0 \leq \theta \leq 2\pi$ Cauchy's integral formula is given by: $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = ...
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2answers
51 views

What does the graph of $5e^{it}$ look like on the complex plane?

I know that $5e^{it} = 5(\cos(t) + i\sin(t))$, but that doesn't really help me. What other information can I use to visualize this graph besides plotting many points and seeing what type of graph it ...
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54 views

Is $pi=ln(-1)/sqrt(-1)$, and if so what does this mean?

Using the complex integral $z=\cos(x)+i\sin(x)$ $\frac{dz}{dx}=-\sin(x)+i\cos(x)$ $dz=i[\cos(x)+i\sin(x)]dx$ $dz=iz\cdot dx$ $\frac{1}{z}dz=i\cdot dx$ $\ln(z)=ix$ $z=e^{ix}$ ...
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1answer
20 views

For fn(z)= 1/nz, If we make fn(0)= 1, does that make the family of functions bounded?

I have a problem that requires me to use a theorem requiring a bounded family of functions. The family provided that I am supposed to use this theorem for is $f_n (z) = \frac 1 {nz}$ when $z \neq 0$ ...
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1answer
38 views

Why is $\int\limits_{\gamma} \frac{1}{z-1} \neq 2\pi i$, $\gamma = \{z : \lvert z \rvert = 1\}$?

$\int\limits_{\gamma} \frac{1}{z-1}$ $\gamma = \{z : \lvert z \rvert = 1\}$ I use Cauchy's integral formula, which says $$\int\limits_{\gamma} \frac{f(z)}{(z-a)^{n+1}} = \frac{2\pi i}{n!} ...
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3answers
24 views

$\int\limits_{\gamma} \frac{1}{z-1}$, $\gamma(\theta) = 2e^{i\theta}$, $0 \leq \theta \leq \frac{\pi}{2}$

$\gamma(\theta) = 2e^{i\theta}$ is a circle centered at $(0,0)$ with radius $2$, so $z = 1$ is inside this path and thus we have to use Cauchy's integral formula for $\int\limits_{\gamma} ...
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1answer
26 views

Evaluate an integral along a semicircle.

Let $\gamma$ be the semicircle $[-R,R]\cup\{z\in\mathbb{C}:|z|=R\ and\ Im{z}>0\}$ traced in the positive direction, and let $R>1$. Evaluate $$\int_\gamma\frac{dz}{(z^2+1)^2}.$$ I want to say ...
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1answer
25 views

Circular Contour Integration .

Doing some revision for an upcoming exam I have stumbled across the following problem: Evaluate the integral $\int_{C}\log(z)$ where $C=C(2,1)$ the positively oriented circular contour, centre 2, ...
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1answer
27 views

Path dependence of integrals

Are the integrals of the function $ \Large f(z)=e^{1/z}dz$ path independent in the domain $D= \{Re z >0\}\setminus\{3\}$?
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3answers
79 views

Finding $\int_0^\infty\frac{\sin^{2}x}{1+x^4}dx$

I am trying to evaluate $$\int_0^\infty\dfrac{\sin^{2}x}{1+x^4}dx$$ and I am stuck on how to start. I am thinking the first step would be to substitute $$\dfrac{(1-e^{2ix})+(1-e^{-2ix})}{4}$$ for ...
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1answer
32 views

Evaluate integral using Residue Theorem

Let $\gamma$ be the semicircle $[-R,R]\cup\{z\in\mathbb{C}:|z|=R\ and\ Im{z}>0\}$, traced in the positive direction, and $R>1$. Evaluate $$\int_\gamma\frac{dz}{z^4+1}.$$ I note that ...
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1answer
26 views

Classify the singularities of the function .

Classify the singularities of the function $\frac{1-\cos(z)}{z^2(z-1)}$. I think my answer may be that I have a simple pole at $z=0$ and a removable singularitie at $z=-1$ however i am not too sure. ...
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1answer
33 views

Prove a union is a domain

Prove that if S and T are domains that have at least one point in common, then S union T is also a domain I wrote: A domain is a set that is open and connected. The union of open sets is easily open. ...
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1answer
55 views

integral of $ \int_{\gamma}e^{1/z}dz$ [on hold]

How do you find the integral of $$ \int_{\gamma}e^{1/z}dz$$ in the domain $ D= \{Re z >0\}$
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22 views

Contour integration with a branch cut. Parameterizing f(z) properly

I have a contour integral of a function of the form $(z^6-P)^\alpha z^\beta$ Here $\alpha\in R$, $\beta\in N$ and $P$ is some constant. I therefore have branch points at the sixth roots of $P$. The ...
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1answer
32 views

Uniform convergence of $\sum^n_{k=-n} \frac{1}{z+k}$

Let $D=\mathbb C \setminus \mathbb Z$ and define $$f_n(z)=\sum^n_{k=-n}\frac{1}{z+k}$$ I have to prove that $\{f_n\}^\infty_{n=0}$ is locally convergent on D. We are given the hint to write $f_n$ as a ...
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0answers
16 views

2nd order pole inwhile computing residue in a complex integral

I was wondering - how does one deal with finding a residue of a contour integral when you introduce a fresh pole while computing the residue. For example: $$ \int \, \frac{ \frac{e^{\sqrt{x^2 + ...
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1answer
44 views

Complex hypersurface globally defined

Let $A$ be a pure one-codimensional analytic subset of a domain $D \subset \mathbb{C}^n$. Is it true that $A$ is defined by one single holomorphic equation $f(z)=0$ if $D$ is bounded and pseudo-convex ...
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1answer
17 views

Quick Question on Zeroes of Transfer Function

Sorry for not providing context here. Suppose I have an output $Y(z)=\frac{z-1}{z}$ and input $X(z)=\frac{z^2+3z+2}{z^2}$ to yield a transfer function ...
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10 views

Prove that an entire function of exponential type is of order at most $1$.

By Entire functions theory, the order (at infinity) of an entire function $f(z)$ is defined using the limit superior as: $$\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln\Vert f \Vert_{\infty, B_r} ...
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0answers
27 views

Existence of certain Analytic functions on the open unit disc ( possible application of Schwarz lemma )

Let $D$ be the open unit disc. Then can there be analytic functions with the property (1) $f(\frac{3}{4})=\frac{3}{4}$ and $f'(\frac{2}{3})=3/4$ 2) $f(\frac{3}{4})= -\frac{3}{4}$ and ...
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1answer
19 views

path dependence of the integral $f(z)=\frac{1}{(z-4)^2} + \sin z$

Are the integrals of the $$f(z)=\frac{1}{(z-4)^2} + \sin z$$ path independent in the following domain $$D= \{\operatorname{Re} z >0\}\setminus\{4\}$$ My thought is that since ...
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2answers
35 views

Entire functions of order 0

Sorry, this may be a stupid question, but I am just beginning to learn about this and cannot find the answer anywhere I have looked so far. Clearly if we have any polynomial $P(z)$, then it is easy to ...
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0answers
38 views

$\tan z=az+b$ has infinitely many solutions

I try to prove following questions. Prove that, for all complex numbers $(a,b)\neq (0,\pm i)$, the equation $\tan z = az + b$ has infinitely many solutions. By assuming $(a,b)=(1,0)$, I tried ...
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1answer
22 views

Bounded entire fuctions. [duplicate]

Let $f$ be an entire function and assume f(0) = 1 and $|f(z)| \large \geq\frac{1}{3}\left| {\LARGE e^{z^{3}}}\right|$ for all $z$. Show $f(z) = {\huge e^{z^{3}}}$ for all $z$. Can this be shown ...
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1answer
49 views

infinite product expansion $\frac{1}{\sin(z)}-\frac{1}{z}$

I have successfully solved that $$\frac{1}{\sin(z)}-\frac{1}{z} = \sum_{n=-\infty, n \neq 0}^\infty (-1)^n\left(\frac{1}{z-n\pi}+\frac{1}{n\pi}\right)$$ and am now attempting to integrate both sides ...
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1answer
29 views

Prove that if integral of f around any closed disk in U is 0 then f is holomorphic in U?

I know goursats theorem says that if integral of f over any triangle in U is 0 then f is locally integrable in U and hence by Moreras theorem is holomorphic in U. But here I need to show that if f is ...
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0answers
22 views

joukowski mapping ellipse to circle and how to find the inverse of this map [on hold]

I have question about the Joukowski map ellipse I know that this map use to map the exterior o the circle to the exterior of the ellipse by using J(z)= ( 1/2 ( r + 1/r ) cost + i 1/2 (r-1/r) sin t ...
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0answers
26 views

Prove that if $g = r +ip$ is analytic on $C$ and $r(x,y) \leq M$, with $M > 0$, for all $(x,y)\in C$, $g$ is constant.

Let $g = r +ip$ be analytic on $C$. If for some $M > 0$ we have $r(x,y) \leq M$ for all of $C$, then $g$ is constant. The theorem is given without proof in my notes and I can't find any examples ...
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1answer
49 views

$f$ is holomorphic iff $df$ is $\Bbb C$-linear

Let $\Omega\subseteq\Bbb C^n$ open connected, $f:\Omega\to\Bbb C$ differentiable in the real sense. We know that $f$ is holomorphic iff $\partial_{\bar z_j}f=0\;\;\forall j=1,\dots,n$ . We know also ...
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0answers
23 views

Locally Lipschitz complex function [on hold]

I want to study the property of being locally Lipschitz for the following function $$f(z)=\vert z\vert^\gamma z^2$$ with $\gamma\in\mathbb{R}$. Some hints to study this problem?
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2answers
27 views

Simply connected domains $D=\{\operatorname{Re}z>0\}\setminus\{1\}$ [on hold]

Is the domain $D=\{\operatorname{Re}z>0\}\setminus\{1\}$ a simply connected domain? How is this shown?