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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A problem related to complex polynomial

Let $$P_{t}(z) =a_{0}(t) + a_{1}(t)z + ...+a_{n}(t)z^n$$ be a polynomial where the coefficients depend continuously on a parameter $t \in (−1, 1)$. Assume that there exists $\text{t}_{0} \in (−1, 1)$ ...
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1answer
46 views

Laurent Series of $f(z)=(z+1)\sin\frac{z^{2}+2z+5}{(z+1)^{2}}$

We are asked to find the Laurent series for the following function. $$f(z)=(z+1)\sin\frac{z^{2}+2z+5}{(z+1)^{2}}$$ Around the point $$z_{0}=-1$$ I have tried to factor the inside of sine, to no ...
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1answer
32 views

Need some help with complex-analysis definitions and understanding

Right, so I'm struggling proving/disproving that for functions $u,v: \mathbb R^2 \to \mathbb R$ if $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (so the relation is ...
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21 views

An $f\in H^{1/2}$ with self-convolution, showing it is an $C^1$ function.

If $f\in H^{\frac{1}{2}}(\mathbb{R})$ is a Sobelev 1/2 function that $f=f*f$, then how do you show that $f\in C^1$ with a bounded derivative.
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2answers
34 views

Contour integrals with $dx$ instead of $dz$

I was wondering whether a contour integral (over a simple, closed contour) changes if we change the differential to only the axis that contains the singularities. Intuitively, I would think there is ...
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1answer
34 views

Complex Integral over an Astroid

The question asks us to compute the complex integral: $$\int \frac{\text dz}{(z^{2}-1)^{2}(z-3)^{2}}$$ Over the positively oriented astroid: $$x^{\frac{2}{3}}+y^{\frac{2}{3}}=2^{\frac{2}{3}}$$ I ...
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54 views

What is the angle at critical point $z=1$ of $\left|z-\frac{i-1}{2}\right|=\frac{\sqrt{5}}{\sqrt{2}}$ under the Joukowski transform?

Question: What is the angle at the critical point $z=1$ of the image of the circle $|z-\frac{i-1}{2}|=\frac{\sqrt{5}}{\sqrt{2}}$ under the Joukowski transform? The Joukowski transform is defined ...
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30 views

If $f$ is entire and $f \circ (\_)^{-1}$ has a pole at $z = 0$, then $f$ is a polynomial.

I was wondering if somebody could help me finish off the proof of this statement; I'm not sure if my approach can be salvaged, but here's what I've got so far: Since $f \circ (\_)^{-1}$ has a pole at ...
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13 views

Some complex variable problem [duplicate]

Problem. Suppose that $f$ has an isolated singularity at the point $a$, and $f'/f$ has a first-order pole at $a$. Prove that $f$ has either a pole or a zero $a$ This is a problem from a past ...
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1answer
72 views

What topics have complex analysis among their prerequisites?

I have one spot left in my bachelor's curriculum and am trying to decide between complex and functional analysis. What the latter is good for, is more or less clear to me: e.g. for advanced ...
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1answer
27 views

Complex matrix calculations

Sorry about the vague subject but I really found some difficulties in calculating complex matrices. Assume $Z$ is a square Hermitian non-singular complex matrix, then we denote $$F= \left[ ...
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1answer
16 views

polynomial residue

Given a complex function f(z), one way to find the residue at a pole is to find the laurent series centered at that pole since the coefficient for the term with exponent of negative 1 is the residue ...
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1answer
60 views

Evaluation of $\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta$ with Cauchy's residue Theorem

I have to proof $$\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta = \frac{2\pi}{3}$$ with Cauchy's residue Theorem. I have showed it, but in my solution, there comes $-\frac{2\pi}{3}$. I Show you ...
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2answers
67 views

What is the difference between a singularity and a pole?

From what I could find, a singularity is a point at which an equation, surface, etc., blows up or becomes degenerate. And a pole of a function is an isolated singular point a of single-valued ...
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1answer
41 views

How can I find the Cauchy Principal Value of this integral using complex analysis?

I'm supposed to solve the real integral using a contour integral (The Cauchy Principal Value). Can someone give me a hand? I cannot seem to be able to do it... This is what I've tried so far: I ...
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1answer
99 views

integration, laurent series, residue therorem

Evaluate the integral $\int_\gamma f(z)dz,$ where $\gamma(t)=e^{it}$, and $0\leqslant t\leqslant2\pi$. For $f(z)$ equal to: $$\dfrac{e^z}{z^3},\quad\dfrac1{z^2\sin z},\quad\tanh ...
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2answers
50 views

Is “being harmonic conjugate” a symmetric relation?

The question is: Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$ are functions and $v$ is a harmonic conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, ...
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1answer
18 views

Existance of an analytic funtion satisfying some condition

Does there exists an analytic function$f:D \to D$ ($D$ is the unit disc) such that $f(\dfrac{i^{n}}{n})=-\dfrac{1}{n^{2}}$?
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20 views

Inequality involving radius of convergence of Taylor series

Let $(a_{n})_{n \in \mathbb{N}} \subset \mathbb{C}$ and $z \in \mathbb{C}$ . Let $f(z)=\sum\limits_{n \in \mathbb{N}} a_{n}z^n$ have radius of convergence $R_{0}$ and let $z_{0}$ be such that $|z_{0}| ...
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1answer
58 views

Existence of Holomorphic function (Application of Schwarz-Lemma)

Let, $D=\{z\in \mathbb C:|z|<1\}$. Which are correct? there exists a holomorphic function $f:D \to D$ with $f(0)=0$ & $f'(0)=2$. there exists a holomorphic function $f:D \to D$ with ...
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1answer
26 views

Prove the following about absolutely convergent complex series

Prove that for every sequence $(a_n)_n$ of complex numbers, if the series $\sum_{n\ge 0} a_n$ is absolutely convergent, then $|\sum_{n\ge 0} a_n| \le \sum_{n \ge 0} |a_n|$. I've been given the ...
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1answer
39 views

Residue theorem with contour integrals

I want to evaluate the integral $$ \int_{\gamma} \frac{1}{z^{2}\sin(z)} dz$$ where $\gamma(t) = e^{it}$ and $ 0 \leq t \leq 2\pi$ using the Residue theorem. I've tried expanding sin(z) with Taylor ...
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12 views

Singularities of complex exponential and asymptotic expansion

Consider the equation $$L[u(x,t)] = \tilde u(s,t) = \frac{e^{-t\sqrt{s^2-1}}}{s-2}$$ I want to find $u(x,t)$ in the form of an integral. I first need to find the poles and singularities of the ...
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1answer
15 views

Question about complex polynomials and derivatives

I have the following problem. Suppose $$ f(z, \overline{z} )= \sum a_{lm} z^l \overline{z}^m$$ is a polynomial. ($z \in \mathbb{C}$). then $f$ contains $\mathbf{no} $ $\mathbf{term}$ with $m > ...
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33 views

2nd order difference equation with n dependent coeff

I wanted to know if there was and solution to the following equation. $\left(N-n+1\right)E_{n+2} - NE_{n+1} + \left(n+1\right)E_{n} + N = 0$ Where $E_0 = 0$ and $E_1 = 2^N - 1$. $N$ is just a ...
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3answers
101 views

Prove exponent(m)=e^{m}

please show me how to do the third one, I just understand the 1st and 2nd, but i have no idea how to do the 3rd. thank you.
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1answer
57 views

bounded interval is bounded and connected

Can you please tell me if my proof is correct? Definition: Let $X$ be a subset of $\mathbb R$. We say that $X$ is connected iff the following property is true: whenever $x, y$ are elements in ...
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24 views

Choosing a line such that a given polynomial does not vanish

Let the polynomial $p(z) = \sum_{j=0}^{n} a_j(2 \pi i z)^n$ where $a_j \in \mathbb{C}$. I need to find a $c \in \mathbb{R}$ such that this polynomial does not vanish on the line $\{z: z=x+ic, x \in ...
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27 views

Application of the mean value theorem to $f(x)=x^{-s}$ (Prop 2.5 in Princeton Lectures in Analysis-Complex Analysis)

The proof of the following proposition in the book says that But I don't quite understand why this is true. Are we using MVT in the Real sense? As far as I am aware of, applying MVT to ...
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1answer
28 views

Show that if $g$ is nonconstant holomorphic and $f$ is harmonic such that $fg$ is harmonic, then $f$ is holomorphic.

Let $\Omega$ be an open and connected set in the complex plane and $g$ be a nonconstant holomorphic function on $\Omega$. Show that if $f$ is harmonic on $\Omega$ such that $fg$ is also harmonic on ...
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21 views

Rouche's theorem to find number of zeroes of slightly perturbed polynomial

Is there a way to use Rouche's theorem to do the following: Say we have $P_{t}(z)$, an $n$-degree polynomial whose coefficients are continuous in $t$, with multiple distinct zeroes at time $t_{0}$, ...
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1answer
53 views

Laurent Series, region of convergence

I want to find the laurent series for $$ f(z) = \frac{z}{z^2 - (1+i)z +i} $$ in powers of $z-1$ and find the region of convergence. I am not quite sure how to do this. I know that $$ f(z) = ...
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3answers
120 views

Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$

Evaluate $$\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\mathrm d\theta$$ This is the final question on my review for my final exam tomorrow, and I will be honest and say that I have no clue ...
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1answer
29 views

Proving Schwartz-lemma-like inequality

I want to show for homework that if $D(0,R)$ is the open disc of radius $R$ centered at $0$ in the complex plane and $f:D(0,R) \to \mathbb{C}$ is holomorphic with $|f(z)| \le M$ for some $M>0$, ...
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0answers
18 views

Proving a bound for $\Gamma(s+u)/\Gamma(s)$

Suppose $s, w$ are complex numbers with positive real part. I have come across a particular bound that I have seen multiple times, but which I do not know how to prove: ...
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50 views

Evaluate the Cauchy Principal Value of $\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2-2x+2)}dx$

Evaluate the Cauchy Principal Value of $\int_{-\infty}^\infty \frac{\sin x}{x(x^2-2x+2)}dx$ so far, i have deduced that there are poles at $z=0$ and $z=1+i$ if using the upper half plane. I am ...
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40 views

Asymptotic expansion of integral (Laguerre)

Consider $$L_n = \frac{1}{2\pi i } \oint_{C'} \frac{1}{(1-t)^{\alpha+1} t^{n+1}} e^{-\frac{xt}{1-t}} dt\,\,\,\,(1)$$ where $C'$ is an anticlockwise contour around zero. Now set $\alpha = n$ and I want ...
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1answer
30 views

Complex number forming equilateral triangle

Suppose we have 3 complex numbers , such that $$|z_1|=|z_2|=|z_3|=1$$ and they form equilateral triangle then will condition $$z_1.z_2.z_3=1$$ always be true? I know cube roots of unity , that is ...
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1answer
28 views

Generating function of the Laguerre Polynomials

The Laguerre Polynomials have the following integral representations $$L_{n}^{\alpha} (x) = x^{-\alpha} e^x \frac{1}{2\pi i } \oint_c \frac{e^{-z} z^{n+\alpha}}{(z-x)^{n+1}} dz$$ where $c$ is an ...
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1answer
51 views

Countour integral using residue theorem

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$. I want to do this using the residue theorem but I am unsure of how to work out the poles of ...
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28 views

Identifying the complex space

How can I identify the complex plane $C^2$ with tuples $(r, z:w)$ where $r \in R_{>0}$ and $(z:w) \in CP^1$ (complex projective line)? Thanks!
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24 views

maximal and minimum degree of laurent polynomial

Let $P$ be a Laurent polynomial. Is there a canonical way to compute its maximum and minimum degree ? Is there a way to separate the positive degrees from the negative ones too ? Thanks.
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1answer
47 views

Is it possible to have Logarithm with base 1 or 0?

I am wondering is there any definition that allows logarithm to have base 0 or 1 in real or complex fields (considering Euclidean space)?? Out-coming question is if you can define a logarithm with ...
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1answer
38 views

Estimate the minimum of a degree 2007 polynomial on circle

Let $f$ be a polynomial of degree 2007: $$f(z)=\sum_{n=0}^{2007} a_n z^n$$ If $f$ has exactly 1966 zeros in the unit disc $D$ of course counting multiplicity, prove that: $$\min_{|z|=1} |f(z)|\leq ...
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1answer
33 views

Evaluating $\int^{\infty }_{-\infty}\frac {z^3\sin az}{z^4+4}dz$

I'd like to evaluate following integral with contour integration $$\int^{\infty }_{-\infty}\dfrac {z^3\sin az}{z^4+4}dz$$ and I think the best way to solve is to recognize it is equal to the ...
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1answer
66 views

Lauren expansion with different annuli

Find the Laurent expansion about $0$ of $$f(z)= \frac{1}{(z-i)(z-2)}$$ on the annuli: $0 \lt \mid z \mid \lt 1 $, $ 1 \lt \mid z \mid \lt 2$, $ 2 \lt \mid z \mid \lt \infty $. So far I have put ...
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1answer
28 views

Upper bound for the modulus of bounded holomorphic function in the unit disk with $f(a)=0$

Let $f$ be a holomorphic on the unit disk and $|f(z)|≤ M$ for $|z|<1$. Assume that $f(a)=0$ for some $|a|<1$. Prove that $|f(z)|≤(M|z-a|)/(|1-a ̅z|)$ for $|z|<1$. Progress I want to apply ...
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28 views

$(1-i)^{(4/5)}$ using De moivre's theorem

I obtained the five roots of $$(1-i)^{(4/5)}$$ and I got: $$z0= 4^{(1/5)}(e)^{(i\pi/5)}$$ $$z1= 4^{(1/5)}(e)^{(i3\pi/5)}$$ $$z2= 4^{(1/5)}(e)^{(i\pi)}$$ $$z3= 4^{(1/5)}(e)^{(i7\pi/5)}$$ $$z4= ...
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2answers
26 views

Entire functions and local injectivity

QUESTION: If a function, $f:\mathbb{C}\rightarrow \mathbb{C}$, is entire and it is non constant is it necessarily locally injective? That is if given some $z_0$, does there exists a disk, ...
2
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0answers
21 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...