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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67 views

How do i write the analytic function $f(z)$ in terms of $z$?

I have an entire function, consider the function : $f(z)= (3x^2 + 2x - 3y^2 - 1) + i(6xy + 2y)$ I want to write $f(z)$ in terms of $z$.
2
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1answer
65 views

Laurent series of $\frac{e^z}{z^2+1}$

I cant figure out the laurent series of the following function. Let $f(z)= \frac{e^z}{z^2+1} $ and $|z|\gt 1$ $$\frac{1}{z^2+1}=\sum_{n=0}^{\infty}(-1)^nz^{-2n-2}$$ and $$e^z = ...
11
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1answer
432 views

Approximate spectral decomposition

A detailed attempt below. I am interested in effective and constructive computations for finding approximate spectral decompositions in some suitable format. Namely, let $A: H \rightarrow H$ be a ...
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0answers
29 views

Laurent Series, How it is done

Suppose that a series $$\sum_{n=-\infty}^{\infty}x[n]z^{-n}$$ converges to analytic function $X(z)$ in some annulus $R_1<|z|<R_2$. That sum $X(z)$ is called the z-transform of $x[n]$ $(n=0,\mp ...
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1answer
36 views

Challanging problems on [Grade-12]Complex Number [on hold]

recently we are introduced to interesting world of complex number but except for 3-5 problems in the my books,all the problems are just plug-and chug,expression manipulation,etc.. which bores me out ...
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1answer
34 views

How can I prove that it is an Entire Function

Prove that if $$ f(z)=\left\{ \begin{array}{ll} \frac{\cos z}{z^2-(\pi /2)^2} & \hbox{when} \; z\neq \mp \pi/2\\ -\frac{1}{\pi}, & \hbox{when} \;z= \pi/2. \end{array} \right. $$ ...
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0answers
29 views

Power series representation with Gamma function

This is taken from Stein and Shakarchi's Complex Analysis (Chapter 6, Exercise 4): Prove that if we take $$f(z) = \frac{1}{(1-z)^\alpha}$$ for $|z|<1$ (defined in terms of the principal branch ...
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0answers
31 views

Non-trivial inverse Laplace transform

I'm trying to compute the inverse Laplace transform of $f(s) = s^c/(N + s^{ir} )$ where $c,N \in \mathbb{C}$ and $r \in \mathbb{R}^+$ using the Bromwich integral $$ F(t) = \frac{1}{2 \pi i} \int_{- ...
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2answers
40 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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2answers
47 views

Is this a legitimate way to compute a contour integral?

I wish to calculate $$\int_{\Gamma}\cos(z)\sin(z)~\text{d}z$$ where $\Gamma$ is the line segment given by $\gamma(t)=\pi t+(1-t)i$ for $0\leq t \leq 1$. Here is what I did: We have that $$\int ...
3
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0answers
38 views

$|p(z)|=1$ contains no circles [duplicate]

Help with the following problem would be appreciated: Let $p(z)$ be a polynomial over $\mathbb{C}$ with at least two distinct roots. Prove that no connected component of the set $\{z \in \mathbb{C} : ...
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0answers
22 views

About the proof of a corollary of Arzela-Ascoli Theorem.

This is from Scheidemann, Complex Analysis. Theorem (Arzela-Ascoli): Let $K$ be a compact separable metric space, $E$ a finite-dimensional Banach space and $(f_j)_{j\in\mathbb{N}}\subseteq C(K,E)$ ...
5
votes
1answer
67 views

Entire function $f$ such that $\lim\limits_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$?

The question is this: Does there exist an entire function $f$ such that $\lim_{z\rightarrow \infty}f(z)=0$ and $f(0)=1$. I immediately would point to $f(z)=e^{-z}$. It is entire and satisfies the ...
0
votes
1answer
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 ...
0
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2answers
49 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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0answers
35 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 ...
0
votes
1answer
26 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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0answers
28 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 ...
2
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1answer
72 views

Analytic continuation of power series on the unit whose terms tends to 0

This problem is from complex analysis. Set $$f(z)=\sum_{n=0}^{\infty}a_nz^n$$ with convergence radius of 1, and $$\lim_{n \to \infty}a_n=0$$ Prove that if $z_0 \in \partial B(0,1)$ is not a singular ...
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1answer
31 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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4answers
132 views

Example 3.40 (b) in Baby Rudin: How to find $\lim_{n\to\infty} \sup \frac{1}{\sqrt[n]{n!}}$?

Here is Theorem 3.39 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Given the power seires $\sum c_n z^n$, put $$\alpha = \lim_{n\to\infty}\sup\sqrt[n]{\vert c_n ...
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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
39 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 ...
0
votes
1answer
25 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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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| ...
0
votes
1answer
57 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 ...
34
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4answers
932 views

Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$

Wikipedia informs me that $$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$ I tried considering $f(x,n) = e^{-x n^2}$ so that its ...
0
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0answers
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= ...
2
votes
2answers
101 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].)
2
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1answer
44 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 ...
0
votes
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 = ...
0
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1answer
37 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 ...
1
vote
1answer
22 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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votes
0answers
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 ...
0
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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 + ...
0
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0answers
20 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
16 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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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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0answers
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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1answer
28 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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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 ...
1
vote
1answer
30 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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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) = ...
0
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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 ...
0
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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 $e^{i\omega_0(t + T)} = e^{i\omega_0t}$ ...
0
votes
1answer
24 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 ...
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2answers
30 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?
2
votes
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 ...
1
vote
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 ...
1
vote
1answer
393 views

Analytic map from punctured disk to punctured plane

If $\mathbb{D}$ is the open unit disk, we know from Liouville's theorem that there cannot exist a 1-1 and onto analytic map (i.e. biholomorphic) from $\mathbb{D}\to\mathbb{C}$? But could there exist ...