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

How to figure out Laurent series expansion for $z^2 \sin(1/(z+i))$?

I am having trouble figuring out the Laurent series expansion for $z^2 \sin\dfrac{1}{z+i}$. I would really appreciate any help.
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0answers
19 views

help solving for $\int \frac{1}{z(1+ln(1+z)}dz$ on path $C := |z|=0.75$ using the residue theorem

given $f(z) = \int \frac{1}{z(1+ln(1+z)}dz$ on path $C := |z|=0.75$ the only singularities inside $C$ are $z = 0$ && $z = \frac{1-e}{e}$ evaluating the residue at $z = \frac{1-e}{e}$ using ...
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3answers
84 views

Are there limits of which we're not able yet to find the value or not even prove non/existence?

I really like working out limits, so I've been wondering: Are there limits we're struggling to evaluate? Are there limits of which we're not succeeding in proving the existence or nonexistence?
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1answer
49 views

Extreme point of unit balls, over $\mathbb C$

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $\mathcal{C}[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
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0answers
60 views

A question about subharmonic functions

On the article about subharmonic functions, wikipedia says In the context of the complex plane, the connection to the convex functions can be realized as well by the fact that a subharmonic ...
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1answer
17 views

how to prove that a given function is univalent

I have to prove that following function is univalent $f(z) = z^2 +3z +1, ~|z|<1$ in complex plane. What I tried is: Let $f(z_1) = f(z_2)$ $\Rightarrow$ ${z_1}^2 +3z_1 +1= {z_2}^2 +3z_2 +1$ ...
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37 views

Does this equations represent the sphere?

Consider complex numbers $x_1$ and $x_2$. Let $j=\sqrt{-1}$. Consider the set of points $$\mathcal{S}=(x_1x_2^*+x_1^*x_2,j(x_1x_2^*-x_1^*x_2,|x_1|^2-|x_2|^2)$$ for each point $(x_1,x_2)$ such that ...
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1answer
25 views

inverse of a point $p$ respect to the circle $|z-z_0 |= r$ in complex

I was solving a problem to find the inverse of a point $p$ respect to the circle $|z-z_0|=r$. In my question I had to find inverse of $1+i$ w.r.t circle $|z+1-2i| = 2$. I applied the formula $q = z_0 ...
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1answer
69 views

Can someone please explain the following definition of $\ln(e^z)$

I noticed someone do this from one of the questions is asked on here i had: $$e^z = -0.5$$ $$e^z = 0.5e^{i\pi}$$ which magically became: $$z = \ln\left(\frac12\right) + iπ + 2ikπ$$ does this mean ...
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1answer
25 views

Showing that the map $f(z) = \frac{1}{z} $ maps circles into circles or lines

Let $f: \mathbb{C} \setminus \{0 \} \to \mathbb{C} \setminus \{0\} $. We want to show that $f(z) = \frac{1}{z}$ maps circles into circles and lines. My professor gave the following hint: The general ...
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1answer
68 views

Dispersion Relations (Mathews and Walker,Mathematical methods of Physics, 2nd edition)

In page no,131 of Mathews and walker, the function if defined as $$F(x)=\frac{P}{i\pi}\int_{-\infty}^{\infty}\frac{F(x')}{x'-x}$$ Then, by using even symmetry $$f(-z)=+f^*(z*)$$ they obtained ...
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2answers
47 views

Studying the complex-valued function $f(z) = \frac{1}{z}$

Let $f(z) = \frac{1}{z}$. I am trying to study this function: First thing to notice is that we can write (after some algebraic manipulation and putting $z = x + i y $) that $$ f(x,y) = \frac{x}{x^2 ...
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1answer
47 views

How to find maximum value?

Let $R$ be the region $z \leq 1$. Compute the maximum value of $|z^2 + z + 2|$ in $R$ and find out the point where this function reaches this value in $R$. I let $$z = \cos \theta + i \sin ...
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0answers
38 views

Extreme point of unit balls, the complex case [duplicate]

I've been trying to determine what are the extreme point of the unit balls of $\ell^1$ and $C[0,1]$. I think that I cracked the real case (I got for $\ell^1$: $\{e_n\}_{n\in \mathbb ...
1
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1answer
31 views

Show that a harmonic function on an open connected set which is holomorphic on some open subset is in fact holomorphic everywhere.

Suppose $f$ is a harmonic function on a connected open set $\Omega$ in the complex plane, and suppose also that $f$ is holomorphic on some open subset $U$ of $\Omega$. Prove that $f$ is holomorphic on ...
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1answer
44 views

The Cauchy Estimate

The Cauchy Estimate is: If a function $f:D\to\mathbb{C}$ is differentiable for $|z-z_0|<R$, $0<r<R$ and $|f(z)|\leqslant M$ for all $|z-z_0|=r$, then $$ ...
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3answers
58 views

Taylor Series of a Complex Function

Consider the function $$ \exp\left(\frac{z}{1-z}\right). $$ Since this is holomorphic for $|z|<1$, then it has a Taylor Series valid for $|z|<1$, i.e., $$ ...
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1answer
44 views

Evaluating $\int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm dx$ [duplicate]

$$\int_{0}^{2\pi } \frac{\sin^{2} (x) }{5+4\cos(x)}\,\mathrm dx$$ I am having trouble parsing the square of sine in the numerator. Could someone provide some hint? Thanks.
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1answer
43 views

Sketching complex numbers on an Argand diagram help

for {${z\in \mathbb C : Im(z)>0}$}, we simply sketch the upper half of the Real axis, right? Then, if we have $z=a+ib$, and we sketch that, and we have $w=iz=-b+ai$ which means $w+1=(1-b) + ai$ ...
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43 views

Help evaluating this seemingly simple integral using residue theorem

$f(z)$ = $\int$$\frac{1}{2e^z+1}$, this is a path integral along $|Z| = 4$ and i know from my textbook that: My attempt: singularity is at: $2e^z = -1$ therefore at $z = ln|0.5|+2ik\pi$ ...
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0answers
32 views

Notion of complex optima

Consider the function: $$y = \frac{1}{3}x^3 + x$$ Suppose we wanted to determine its local optima, but instead of looking at local optima with domain $R$ we instead consider domain $C$ and range ...
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1answer
65 views

Contour integral $\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz$ using residues

I am trying to evaluate the contour integral $$\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz.$$ In this case the two singular points lie on the boundary (on the contour). So do I count the residues at this ...
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2answers
48 views

Cauchy integral theorem for star-shaped regions vs Cauchy integral theorem

I don't get the point of Cauchy integral theorem for star-shaped regions. Doesn't "normal" Cauchy integral theorem imply this? Why some books incorporate just Cauchy integral theorem for star-shaped ...
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0answers
26 views

Expressing the zeta function in another way (Problem 1 in Ch.6 Princeton Lectures in Analysis-Complex Analysis)

I cannot get the term $ -\frac{N^{s-1}}{s-1} $. My attempt $$\begin{align} \zeta(s) &= \sum^{\infty}_{n=1}\delta_n(s) + \frac{1}{s-1}\\ &=\frac{1}{s-1} + \sum_{1\leq n\leq N}\delta_n(s) ...
3
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1answer
65 views

Approximating $L^p$ functions using Schwartz functions with compact support on the Fourier side

For $1\leq p<\infty$, how would you show for any $f\in L^p(\mathbb{R})$ and given $\epsilon>0$, there exists $L<\infty$ and $g\in \mathcal{S}(\mathbb{R})$ such that $\|f-g\|_p<\epsilon$ ...
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44 views

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
59 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
39 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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0answers
28 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
38 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
38 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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0answers
59 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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35 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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0answers
14 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
32 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
25 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
82 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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79 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
54 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
109 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
62 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
27 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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25 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
98 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
35 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
54 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
21 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
23 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 > ...
1
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0answers
39 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 ...
1
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3answers
122 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.