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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22 views

$4^{th}$ root of $-6i$

I want to find the $4^{th}$ roots of $-6i$. What I do is: $$z^4 = -6i$$ $$z^4 = r^4 e^{4i\theta} = 6e^{-i\frac\pi2}$$ $$\implies r=\pm 6^{\frac14}, 4\theta = -\frac\pi2\implies \theta ...
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4answers
34 views

Simplifying$\left|\frac{z-3}{z+3} \right|=2$

I want to graph the following, but simplifying is the question here: $$\left|\frac{z-3}{z+3} \right|=2$$ Now I can do this : $$\frac{|z-3|}{|z+3|}=2 $$ $$|z-3|=2|z+3|$$ $$|x+iy-3|=2|x+iy+3|$$ What ...
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2answers
19 views

Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin. $f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad ...
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3answers
29 views

How is this step done? $\left|\frac{i\overline{z}}{2} -\frac i2\right|=\frac{|z-1|}{2}$

Absolutely everything makes sense other than what is in red. How is this step completed? Let us show that if $f(z)=\dfrac{i\overline{z}}{2}$ in the open disk $|z|\lt 1$, then$$\lim ...
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4answers
30 views

differentiability, complex analysis [on hold]

I've been looking at this and have no idea where to start or how to solve this
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1answer
23 views

Contour integration

Consider the real-valued function $$u(t) = \frac{1}{13-12\cos(t)}$$ By converting it to a contour integral along the unit circle in $\mathbb{C}$, evaluate $$\int_0^{2\pi} u(t)\;dt$$ I have ...
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1answer
34 views

Liouville's theorem application

Suppose that $f(x+iy) = u(x,y) +iv(x,y)$ is differentiable on $\mathbb{C}$ and $u$ is bounded on $\mathbb{C}$. Use Liouville's theorem to show that $f$ is constant on $\mathbb{C}$. Hint: what ...
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1answer
27 views

Functions of one complex variables [on hold]

let $G = C \setminus\{0\}$ and show that every closed curve in $G$ is homotopic to closed curve whose trace is contained in $\{\,z : |z | = 1\,\}$.
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1answer
38 views

The existence of anti-derivatives

The only thing I can think of is that the function is continuous hence the anti derivative exists. I was wondering if there is anything else that needs to be done/said?
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1answer
15 views

Complex power series which converges absolutely on the boundary converges absolutely on a neighborhood of the boundary

If a complex power series $\sum_{n = 0}^{\infty} a_n z^n$ converges absolutely for $|z| \leq 1$, does it necessarily converge absolutely for $|z| < 1 + \epsilon$, for some $\epsilon > 0$?
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2answers
26 views

Cauchy Riemann equation

I've done part (a) as it simply involves stating the Cauchy Riemann, however part (b), im assuming I would have to apply this and therefore for part (i) let $u'=(u^2-v^2)$ and $v'=2uv$ would this ...
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0answers
25 views

one complex variable conway

Please I want some help for "exercise 2 page 99 in conway second edition". Let G be open and suppose that $\gamma$ is rectifiable curve in G: $\gamma$ ~ $0$. Set $r=d(\gamma , G)$ and $H={Z \in C , ...
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0answers
49 views

Integrability problem in Cauchy Integral Formula

This is problem on the integrability of a 2-form in the nhbd of its singularity. I was looking at the general Cauchy formula (general because it works for $\mathcal C^1$ function, and makes the case ...
1
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1answer
9 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
19 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
35 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?
2
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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
16 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
20 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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1answer
19 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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1answer
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
19 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
23 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
55 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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2answers
225 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
30 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$Arg\ w=\pi/2$ ...
2
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1answer
47 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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0answers
39 views

Computing an integral using residues

I am trying to find an integral: $$\int_{-\infty}^{+\infty}\frac{e^{-\sqrt{(x^2 + 1)}}}{(x^2 + 1)^2}\,\mathrm dx$$ I went about applying contour integral over a semicircle with diameter along $ x = ...
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1answer
26 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
52 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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votes
1answer
57 views

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

Using the complex integral $z=\cos(x)+i\sin(x)$ $\frac{dz}{dx}=-\sin(x)+i\cos(x)$ $dz=i[\cos(x)+i\sin(x)]dx$ $dz=iz\cdot dx$ $\frac{1}{z}dz=i\cdot dx$ $\ln(z)=ix$ $z=e^{ix}$ ...
2
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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 $z \neq 0$ ...
1
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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!} ...
0
votes
3answers
26 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} ...
0
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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 ...
1
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1answer
28 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
81 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
33 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 ...
2
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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
56 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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0answers
23 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
33 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 ...
0
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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 + ...
1
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1answer
47 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
18 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 ...