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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Prove locally uniformly convergence of a sequence

Let for $n = 0,1,2,...$ , $f_n : [0,1] \rightarrow \mathbb{R}$ defined by $f_n (x) = x^n$. 1) Is the convergence of {$f_n$}$_{n=0} ^\infty$ to $f$ locally uniformly on the interval $[0.1]$? 2) And ...
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11 views

Hodge star/ Technical question Extended

I wanted to maybe extend Hodge star/ Technical question to a new question so others could benefit from the idea. So there we discussed that when the $\star$ is Hodge duality star then it is ...
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1answer
17 views

Modular forms: What is $\mathbb{H} / SL_2(\mathbb{Z})$?

I am beginning to understand the very basics of modular forms, in that I understand the concept of a weakly modular function, I have seen the examples of $G_k(z)$ and $E_k(z)$ as weakly modular ...
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1answer
15 views

Two holomorphic functions which have a simple roots at the origin

I am trying to solve the following question: Let $f$ and $g$ be functions holomorphic on the closed unit disk. Assume that f and g have simple zeros at the origin and that g has no any other root in ...
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1answer
21 views

Cauchy formula in Polydisks

I don't understand a remark after the proof. Here's the theorem: The proof is done by induction on $n$; starting from $n=1$ on the unitary disk in $\Bbb C$, which is the well known Cauchy formula. ...
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14 views

contour integral

I am looking for help in find the contour integrals of I want to know what is a good theorem to use in this integral I do not know who to deal with power 1/3 and 2/3 when I need to find the ...
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1answer
19 views

Radius of convergence of a complex function with a Taylor Series expansion

The function $$f(z)=\frac{1}{1+i-\sqrt{2}z}$$ has a Taylor series expansion around $z_0=0$. What is its radius of convergence? So far I have computed the singularity point to be $z = ...
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13 views

Proof of Contour Integrals and Limits

I've been thinking about the following proof and I'm simply not sure where to start, so any help is appreciated. Thanks in advance. Proof: Let $E$ be a domain in $\mathbb{C}$ and let $f$ be an ...
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3answers
172 views

Geometry with complex numbers.

Let $a$, $b$, $c$, and $d$ be four complex numbers on the unit circle, such that the line joining $a$ and $b$ is perpendicular to the line joining $c$ and $d$. Find a simple expression for $d$ in ...
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16 views

Find the image of the set $\{z \in \mathbb C | -3\lt Re(z) \lt 5, -1 \lt Im(z) \lt 6 \}$, under the function $e^z$

Find the image of the set $\{z \in \mathbb C | -3\lt Re(z) \lt 5, -1 \lt Im(z) \lt 6 \}$, under the function $e^z$ So I know that I should like it as $e^z=e^{x+iy}=e^x(\cos(y)+i \sin(y))$. And I ...
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16 views

holomorphic function writen as a serie

To each complexe number $\alpha\in \mathbb{C}$ we associate a function defined by: $$ f_\alpha(z)=1+\sum_{n\geq 1}\frac{\alpha(\alpha-1)...(\alpha-n+1)}{n!}z^n $$ I want to show that this function is ...
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1answer
17 views

A little guidance on finding the limit

How do I find the limit of $f(z) = \frac{x^2y}{x^3+y^3} + ixy$ as $z \to0$ ? What I think is if $z\to0$, that implies $x ,y\to0$. But since the $f(z)$ has both variables $x$ and $y$ mixed together, ...
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2answers
19 views

All solutions of $z^2 + (i-2)z + 3-i =0$

I want to find all solutions of $$z^2 + (i-2)z + 3-i =0$$ Now this is what I do: $$x^2 - y^2 +2xyi +(i-2)(x+iy)+3-i =0$$ $$x^2-y^2 +2xyi + xi-y-2x-2iy+3-i=0$$ $$x^2-y^2-y-2x+3+i(2xy+x-2y-1)=0$$ Now ...
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33 views

Finding roots of $z^3 = 8$

I am having trouble finding the cubed roots of $8$ as a complex number. $$z^3 = 8+0i$$ $$z^3 = r^3 e^{3\theta i}=8e^{2i\pi k},\quad k\in \Bbb Z$$ $$\implies r=2,3\theta = 2\pi k\implies \theta = ...
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1answer
13 views

Upper bound for Estimation Lemma

I am struggling with the following question using the Estimation Lemma: Let $ \gamma$ describe the semi-circle $Re^{it}$, where $ 0 \le t \le \pi$, and $ R \gt 3$. Show that $$\int_\gamma ...
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38 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
43 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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21 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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33 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
36 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
25 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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29 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
16 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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26 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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54 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 ...
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1answer
11 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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21 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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36 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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23 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
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
22 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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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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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
58 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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227 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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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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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
57 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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41 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
29 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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21 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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53 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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1answer
57 views

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

Using the complex integral $z=\cos(x)+i\sin(x)$ $\frac{dz}{dx}=-\sin(x)+i\cos(x)$ $dz=i[\cos(x)+i\sin(x)]dx$ $dz=iz\cdot dx$ $\frac{1}{z}dz=i\cdot dx$ $\ln(z)=ix$ $z=e^{ix}$ ...
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1answer
24 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 $z \neq 0$ ...