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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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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0answers
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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0answers
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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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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0answers
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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0answers
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
52 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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0answers
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
29 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 ...
2
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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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0answers
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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0answers
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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0answers
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, ...
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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 $0< b < 1$. ...
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1answer
43 views

Compute $f''(0)$ for a holomorphic function on a square given $f'(0)$ and $f(0)$

Let $S$ be the square $\{x + iy: |x| < 1, |y| < 1\}$ and $f:S \rightarrow S$ a holomorphic function so that $f(0)= 0$ and $f'(0) = 1$. Find $f''(0)$. It seems like I need to use Cauchy's ...
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4answers
102 views

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$ I wasnt exactly sure how to approach this. I saw some similar examples that used Cauchy's theorem.
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2answers
35 views

Poles of $\frac{1}{1+x^4}$

The integral I'd like to solve with contour integration is $\int^{\infty }_{0}\dfrac {dx}{x^{4}+1}$ and I believe the simplest way to do it is using the residue theorem. I know the integrand has four ...
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1answer
23 views

Let $f(z)=(z^3-1)^{1/2}$, find a branch of the logarithm that makes $f(z)$ holomorphic inside the unit disc and satsfies $f(0)=i$

Let $f(z)=(z^3-1)^{1/2}$, find a branch of the logarithm that makes $f(z)$ holomorphic inside the unit disc and satsfies $f(0)=i$ I tried factoring out a $z^3$ and writing ...
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0answers
8 views

Proving susceptibility in Lorentz Model satisy Kramers-Kronig relations

My instructor asked me to prove that the real and imaginary parts of the electric susceptibility derived from Lorentz Model satisfy the Kramers-Kronig relations using the residue theorem. The problem ...
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1answer
20 views

The Convergence of a Complex Valued Infinite Series

Check the convergence and calculate the radius of convergence of the series $$ \sum^{\infty}_{n=1}\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}z^n,\forall\alpha\in\mathbb{C}. $$ I tried to use the ...
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1answer
47 views

Mobius transformation that preservers the unit circle

How do I show that each mobius transformation that preservers the open unit circle (maps it to itself) must be of the form: $c \frac{z-z_0}{\bar{z_0}z-1}, |c|=1, |z_0|<1$ ? I've seen previous ...
2
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3answers
40 views

How to prove $\lim_{x\to \infty}\Im\left(xi^{1/x}\right)=\frac{\pi}{2}$?

I have yet to study complex analysis, but I sperimentally found $$\lim_{x\to \infty}\Im\left(xi^{1/x}\right)=\frac{\pi}{2}.$$ W|A agrees with me too, and while I know that's not so significant, ...
1
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1answer
38 views

Entire function that is strictly increasing on the real line

STATEMENT: Let $f$ be an entire function which maps the real line into the real line and the upper half-plane into the upper half-plane. Prove that $f$ is strictly increasing on the real line. ...