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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24 views

How to show $f(z)=x^2+y^2+i2xy$ is differentiable at $z_0=x_0+i0$?

How to show $f(z)=x^2+y^2+i2xy$ is differentiable at $z=x_0+i0$? Here is what I have done we know by the Cauchy Riemann (its it very easy to verify) that these can only hold for $z_0=x_0+i0$ that is ...
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34 views

Radius of convergence of the series $\sum\limits_{-\infty}^{\infty}(2^{-n}+4^{-n}) z^n$

I'm trying to find for what values of $z\in\mathbb{C}$ the series $$\sum_{n=-\infty}^{\infty}(2^{-n}+4^{-n})z^n$$ converges. My main methods are the nth root test and the ratio test. I believe it can ...
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27 views

When finding Laurent series when to use partial fractions?

When finding the Laurent series of $$f(z):=\frac{1}{z(z-1)(z-2)}$$ valid in the region $1<|z-2|<2$ for example do we just use partial fractions to break $f(z)$ up and the just find the Laurent ...
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23 views

Evaluating an integral using Cauchy Integral Formula and a further application

Question: $i)$ Evaluate $$\int_{\gamma}\frac{e^{2z}}{z}dz$$ Where $\gamma=${$z\in \Bbb{C}: \lvert z\rvert$=1} $ii)$ Hence find $$\int_{0}^{2\pi}{e^{2\cos(t)}}.\cos(2\sin(t) dt$$ My attempt: $i)$ ...
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2answers
48 views

Holomorphic function with $f(z)^2=z$

Is there an holomorphic function $f:B_1(0)\setminus\{0\}\rightarrow\mathbb{C}$ with $f(z)^2=z$?
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1answer
30 views

Help in this inequality in Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 118 he write the following inequality: Why is this inequality true?
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1answer
23 views

Why $|\alpha|\lt 1$ and $|\beta| \gt 1$?

I'm reading Conway's complex analysis book and on page 117 he writes: I didn't understand why $|\alpha|\lt 1$ and $|\beta| \gt 1$. I could only prove $\beta\lt -1$.
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1answer
17 views

disk of convergence for complex-valued series

Find the disk of convergence of $\displaystyle \sum_{k=0}^{\infty} \frac{(z+2)^k}{(k+2)^3 4^{k+1}}$, where $z \in \mathbb{C}$. I tried applying the ratio test: $\lim_{k \to \infty} \left| ...
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1answer
56 views

Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
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2answers
29 views

Geometrical Description of $ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2} $

The question is in an Argand Diagram, $P$ is a point represented by the complex number. Give a geometrical description of the locus of $P$ as $z$ satisfies the equation: $$ ...
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22 views

Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
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22 views

Series expansion of complex exponential

Prove that $e^z= \displaystyle \sum_{k=0}^{\infty} \frac{z^k}{k}$. I took the taylor series $f(z)=\displaystyle \sum_k \frac{f^{(k)}(z_0)}{k!}(z-z_0)^k$ centered at $z_0=0$ and obtained $$e^z= ...
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2answers
99 views

Integral of $e^{\cos t}$

I’d like help with computing the following integral: $$\int_0^\pi e^{\cos t}\,dt.$$ (This is a problem in complex analysis [supposedly].)
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1answer
29 views

Absolute converge of real and complex parts of a series

If the real and imaginary parts of a complex series converge absolutely, then the complex series converges absolutely. Is this true? If we write our complex series $\sum_{k=0}^{\infty} b_k = ...
2
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1answer
40 views

How to find the Laurent series of $\frac{1}{z^4(1-z)^2}$ for |z|>1?

A hint is given that $$\frac{1}{(1-\frac{1}{z})^2} = \frac{z^2}{(1-z)^2}$$ and we know that $$\frac{1}{1-w} = \sum_{n=0}^{\infty} w^n$$ for $|w|<1$. I don't know how to make ...
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26 views

How to calculate this integral using Residues?

$$\int_{-\infty}^{\infty}\frac{\sin(x)}{x^2-2x+5}dx$$ I have calculated the Residue in the upper half plane to be $1/4i$ which is correct according to wolfram alpha but I am unsure on how to proceed I ...
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19 views

Question regarding the Cauchy Residue Theorem

Sorry about the vague title, I'm not quite sure how to word it. Any edits would be very helpful! Question: Let $f(z)$ be analytic with $f'(z_0)\ne0$ where $z_0$ is a complex number Define $$g(w)= ...
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0answers
15 views

calculating $\int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2}$ using Residual Theorem [duplicate]

Could anyone help me provide a way to calculate $$ \int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2} $$ using the Residue theorem in complex analysis? Many thanks
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2answers
23 views

First two terms of the Taylor series of the $n$-th iterated of a holomorpic function

Let $G$ be a region in $\mathbb{C}$ (i.e. $G ≠ \emptyset$ is simply connected and open), with $0 \in G$. Let $f: G \to G$ be a holomorphic function that's Taylor series (around $0$) has the shape $z + ...
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22 views

Name or reference about a inequality with integrals?

I have wrote down some class notes and I think I copied something wrong. It is an integral inequality; $$\iiint_{B^n}|\nabla\psi|^2\frac{1}{|x|^{n-2}}dV\leq C\iint_{\partial B^n}|\psi|^2dA$$ where ...
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0answers
32 views

Help is these inequalities in Conway's complex analysis book true?

I'm reading Conway's complex analysis book and on page 116 he writes: I have two questions: I didn't understand why there is such $M$. What I know is if $f(z)$ has a removable singularity at ...
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1answer
21 views

Littlewood polynomial and Gutzmer-Parseval inequality

Consider the set of Littlewood polynomial for $n \geq 1$, ie $$ L_n = \left \{ a_0 + a_1 z + \cdots + a_n z^n: \quad a_j = \pm 1 \right \} $$ By Gutzmer-Parseval inequality, for some $f \in L_n$, we ...
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1answer
29 views

A holomorphic function $f:D\rightarrow D$ such that $\mid f(z^2)\mid\geq \mid f(z)\mid$ for all $z$ must be constant.

I ran into this reading some Complex stuff for fluid dynamics, and it seems so simple but it's got me stuck. $D$ stands for the unit disk. Since $\mid z \mid<1$, then $\mid z^2\mid \leq \mid z ...
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1answer
32 views

Prove $f(z_0)I(\gamma;z_0)=\frac {g'(z_0)}{2\pi i}\int_{\gamma} \frac {f(z)}{g(z)-g(z_0)}dz. $

Let $f(z)$ and $g(z)$ be analytic in a region A and let $g'(z) \neq 0$ for all $z \in A$. Let g(z) be one to one and let $\gamma$ be a closed curve in A. Show that $$ f(z_0)I(\gamma;z_0)=\frac ...
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1answer
34 views

Why does this $R_0$ exist?

I'm reading Conway's complex analysis book and on page 116 he writes: I didn't understand why such $R_0$ exists.
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1answer
36 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...
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0answers
11 views

name/term for the property of non-analytic complex functions causing “anisotropy”

I'm looking for a mathematical term here so I can understand the consequences of nonlinearity in a system of interest to me. Here's an example system that exhibits this behavior: $$ f(z) = ...
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3answers
49 views

How do I express $f(z)= \frac{6z}{z^2 - 4z + 13}$ as a power series centered at 0?

I am having trouble solving this power series problem because I usually go about decomposing the $f(z)$ and then using geometric series, but the method doesn't seem to work with this because I get ...
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2answers
29 views

$|\exp(iRe^{i\theta})|=\exp(-R\sin\theta)$?

I'm reading Conway's complex analysis book and on page 116 he writes: Why is the last equality true?
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1answer
22 views

Confusion concerning the Sokhotski–Plemelj theorem: two different values for the same real integral

A very well-known formula in complex analysis is $ \lim_{\epsilon\to0^+}\int_{-\infty}^\infty\frac{f(x)}{x-x_0\pm i\epsilon}dx = P\int_{-\infty}^\infty \frac{f(x)}{x-x_0}dx \mp i\pi f(x_0), $ known ...
2
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1answer
43 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
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1answer
27 views

Proof of the linearity of complex integrals for paths of bounded variation?

I am familiar with the proof of the linearity of complex integrals for piece-wise smooth paths. Nonetheless, complex integrals can be defined for more general paths $\gamma:[a,b]\to\mathbb{C}$ where ...
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2answers
22 views

expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
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1answer
31 views

Using Cauchy Integral Formula to solve an integral

Question: Evaluate $$\int_\Gamma \frac{\sin(z)}{(z-\pi)^2} dz$$ Where $\Gamma$ consists of the sides of the rectangle with vertices at $(1,\pm3i)$ and $(4,\pm2i)$ My attempt: The only ...
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0answers
51 views

Why is the fundamental period $T_0$ of the complex exponential $e^{i\omega_0t}$, $T_0 = \frac{2 \pi}{|\omega_0|}$?

Assuming that $\omega_0 \in \mathbb{R}, t \in \mathbb{R}, T \in \mathbb{R}$. I realize that in order for $e^{i\omega_0t}$ to be perioric, it must be true that $e^{i\omega_0(t + T)} = e^{i\omega_0t}$ ...
2
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1answer
45 views

Where is my mistake calculating $\int_{-\infty}^{\infty}\frac{x\sin(x)}{x^2+4}~dx$?

Where is my mistake calculating $$\int_{-\infty}^{\infty}\frac{x\sin(x)}{x^2+4}~\text{d}x$$ Let $$f(z)=\frac{z\sin(z)}{z^2+4}$$ it has simple poles at $\pm 2i$. We take the standard half circle path ...
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1answer
31 views

Can the Heaviside step function be extended for complex values?

Title says it all. In order to apply Cauchy's theorem I need a to extend the step function. So, can the Heaviside step function be extended for complex values, such that it is holomorphic except at ...
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27 views

How to prove the interior and function?? [on hold]

How do I prove that a function $f:D\to C$ is continuous iff $f^{-1}(\overset{\circ}{A})$ is a subset of the interior of $f^{-1}(A)$ for all $A\subseteq C$, where $\overset{\circ}{A}$ denotes the ...
2
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1answer
41 views

Typo in Conway's complex analysis book

I'm reading Conway's complex analysis book and on page 114 he wanted to prove that $$\int_{-\infty}^{\infty} \frac{x^2}{1+x^4}dx=\frac{\pi}{\sqrt2}$$ In order to demonstrate this fact he find ...
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0answers
27 views

Product of integrable complex functions integrable?

Let $f:U\to \mathbb{C}$ and $g:U\to \mathbb{C}$ be two complex functions defined on an open subset $U\subset \mathbb{C}$. My question: If there are holomorphic functions $F:U\to \mathbb{C}$ and ...
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0answers
32 views

The singular points and residues of $\sin(\frac 1 z)$

I met a question asking all the singular points and corresponding residues of $$ \sin \frac 1 z $$ My understanding is that $$\sin \frac 1 z=\frac 1 z-\frac 1{3!z^3}+\frac 1 {5!z^5}+... $$ Thus ...
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0answers
23 views

A complex sequence of functions $(f_n)$ is continuously convergent iff it's compactly convergent against a continuous function

Let $G \subseteq \mathbb{C}$ be a region in $\mathbb{C}$, i.e. $G$ is open, nonempty and connected, and let $f_n: G \to \mathbb{C}$ be a sequence of complex-valued functions, with $n \in \mathbb{N}$. ...
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35 views

Complex Roots (Numerical Methods)

I was given the following question in my Numerical Method exam and I think it is related to Newton's Basis Polynomial, but couldn't solve it. Could anyone guide me to the solution? Show that for ...
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0answers
55 views

Explaining why a complex integral is equal to 0

Question: $$f(z)= \frac{4z}{z^2-1}$$ It has singularities at $z_0=1 \ \text{and} \ z_0 = -1$ with $\operatorname{Res}(f,1)=2 \ \text{and} \operatorname{Res}(f,-1)=2$ Explain why ...
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1answer
45 views

How to classify singularities and calculate residues of $f(z)=\frac{(\cos(z)-1)\sin(z)}{e^{3z}z^4(z-\pi)^2}$

How to classify singularities and calculate residues of $$f(z)=\frac{(\cos(z)-1)\sin(z)}{e^{3z}z^4(z-\pi)^2}$$ I have found the singularities to be $z=0,\pi$. They are both isolated. I first ...
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2answers
31 views

Find the radius of convergence with a function including log

Question: Find the radius of convergence for $$\sum_{n=2}^\infty \frac{\log(n)}{n.2^n}.z^n$$ My attempt: Edit: After the comments I have noticed I have made many errors, here is my updated ...
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1answer
54 views

Show using the definition, that f is differentiable at $x_0+i0$

Question: Show using the definition, that f is differentiable at $x_0+i0$ where $$f= x^2+y^2+2xyi$$ My attempt: I know I must use the definition of differentiability but I cannot see where ...
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0answers
7 views

Representation formualae for analytic functions on $\mathbb{C}^+$ which are not of Herglotz-type

I am working on a problem involving wave equations with dispersions. Quite naturally, these equations involve functions $z \mapsto f(z) \in \mathrm{Mat}_{\mathbb{C}}(N)$ with domain $\mathcal{D}$ and ...
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0answers
39 views

Complex Matrix Representation

Lets say if $X\in C ^{m\times n}$, it does have real and imaginary parts. If I want to represent a matrix in real and imaginary form then why I write it this way where is $i$? \begin{bmatrix} X_r ...
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1answer
21 views

Residue of a function ith different poles

I don't understand the part I have highlighted in green.