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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29 views

Theoretical question regarding Cauchy integral theorem

As we know, according to the Cauchy integral theorem we can easily evaluate an integral of an analytic complex function along the curve connecting two points in a complex plane. Thus for a curve ...
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162 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
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0answers
35 views

Exercise on maximum modulus principle.

"Suppose that $\Gamma$ is the boundary of the unbounded region $\Omega$, in the complex plane $\mathbb{C}$, $f$ is holomorphic in $\Omega$, continuous in $\Omega \cup \Gamma$, and there exist ...
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2answers
60 views

Analytical continuation of $F(p) = \sum_{n \neq 0, n \in \mathbb{Z}} \frac{e^{ipn}}{\sinh^2\kappa n}$

I am trying to find out the behaviour of the series $$ F(p) = \sum_{n \neq 0, n \in \mathbb{Z}} \frac{e^{ipn}}{\sinh^2\kappa n} $$ under analytical continuation in the complex $p$-plane ($\kappa$ is ...
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75 views

Holomrphic function with $f\circ f=f$

I have to solve this problem but i have no idea how to do ...Thanks! Let be $f:\mathbb{C}\rightarrow \mathbb{C}$ a holomorphic function. Show that if $f\circ f=f$ then f is either constant or $f(z)=z ...
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1answer
20 views

Finding the convergence radius of a complex laurent series

Find the maximal ring where the following series converges: $$\sum_{n=1}^\infty\frac{3^n+2^n}{(z-5)^n}+\sum_{n=0}^{\infty}\frac{n^2}{20^n}(z-5)^{2n}$$ I think that taking the minimum between the ...
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2answers
55 views

How I could evaluate this :$ \sum_{n=1}^{\infty}({-1})^{n+1}n(tan^{-1}s-s+\frac{s^3}{3}+…({-1})^{n+1} \frac{s^{2n+1}}{2n+1}) $?

let $s$ be a complex variable which $Re(s)>0$. Evaluate : $ \sum_{n=1}^{\infty}({-1})^{n+1}n(tan^{-1}s-s+\frac{s^3}{3}+....({-1})^{n+1} \frac{s^{2n+1}}{2n+1}) $ I would be interest for any ...
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1answer
36 views

continuous branch of an analytic function

For which real $\alpha,\beta>0$ is there a single valued branch $f$ of the analytic function $f(z)=z^{\alpha}(1-z)^{\beta}$ such that $f$ is defined on $\mathbb{C}\backslash[0,1]?$ I know that the ...
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1answer
57 views

Rouches Theorem Applied to a family of Polynomials

I would like to prove that the family of polynomials $z^{2j+2} + \alpha z^{2j+1} - \alpha z - 1$ has only one root inside the open unit circle when $|\alpha|$ is greater than 1. This seems like an ...
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39 views

How to compute the limit of a function of two variables?

In order to check if $$\;f(z) = \begin{cases} {(\overline z)^2\over z} = {x^3-3xy^2 \over x^2 + y^2}+i{y^3-3x^2y \over x^2 + y^2}, & z \neq 0\\ 0, &z=0 \end{cases} $$ is differentiable at ...
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0answers
12 views

Taylor's expansion of the singular part of an analytic function

Assume $f$ is analytic on the annulus $R_1<|z-a|<R_2$. Assume $R_1<r<|z-a|$. Define $f_2$ by $$f_2(z)=\frac1{2\pi i}\int_{|x-a|=r}\frac{f(x)dx}{x-z}$$ $f_2$ is analytic on $|z-a|>r$. ...
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1answer
51 views

Is an algebra the smallest one generated by a certain subset of it?

Let $X$ be a completely regular topological space and let $BC(X)$ denote the space of bounded continuous complex-valued functions on it. Also, let $C(X,[0,1])$ be the set of continuous functions on ...
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1answer
60 views

Radius of convergence of $\sqrt{\sin(z)/z}$

I'm trying find an analytic definition for $f(z)=\sqrt{z \sin(z)}$ around $0$ ( i.e. $f$ such that $f^2(x) = z \sin(z)$ .) If I write $z \sin(z) = z^2 \frac{\sin(z)}{z}$ then I may take the definition ...
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1answer
45 views

If $f$ is holomorphic in $B(0,1)$ and $|f|>2$ on the boundary, then its range contains $B(0,2)$

Let $f\in Hol\overline{\big(B(0,1)\big)}$ and assume $f(0)=1$ and $\forall z: |z|=1\Rightarrow|f(z)|>2$. Prove that $B(0,2)\subset f\big(B(0,1)\big)$ using the argument principle. I thought ...
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72 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
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2answers
151 views

Gaussian integral with offset, and other cases

Consider the Gaussian Integral $$ \int_{-\infty}^{\infty} e^{-x^2} \ dx = \sqrt{\pi}$$ Numerically, it seems that for any arbitrary imaginary offset, ki, $$\int_{ki-\infty}^{ki+\infty} e^{-x^2} \ dx ...
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2answers
38 views

solving $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}|$ graphicly [closed]

how to solve $|(x-3)(x-1)| $$\le$ $|\frac{1-x}{x-3}| $ in the graphic method?
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0answers
26 views

How to calculate $\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$ using Cauchy's integral theorem? [duplicate]

I need a hint. Where do I start if I want to calculate $$\int_{-\infty}^\infty e^{-t^2/2}\cos2t\ dt$$ using Cauchy's integral theorem?
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4answers
55 views

Can an analytic function take a simply connected region to a non simply connected region?

Let $f$ be a function who is analytic $\Omega \rightarrow \mathbb{C}$ and let $R$ be a simply connected open region in $\Omega$. Is $f(R)$ simply connected?
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31 views

Using a branch cut to integrate a contour

I came across this homework question and I have no clue how to approach it. \begin{align}&\text{Let}\quad {\rm f}\left(z\right)= \frac{z^2 + 3}{\left(z^{2} - 1\right)\left(z - 1\right)} ...
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1answer
59 views

Interesting examples of Cauchy's Integral formula [closed]

Question : What are some interesting and, albeit, counter intuitive examples of real integrals that are solved using Cauchy's integral Formula. Cauchy integral formula can magically transform some ...
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0answers
26 views

How to get between the equivalent definitions of Newtonian capacity?

Here are the two definitions: (1) $$\mathrm{Cap}(A)=\left[\inf \left\{\int\int |x-y|^{d-2}\mu(dx)\mu(dy):\mu \; \text{a probability measure on} \; A \right\}\right]^{-1}$$ and (2) ...
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2answers
58 views

Integral of the conjugate of a complex function

Let $f(z)$ be analytic on some domain containing the closed unit disk $|z|\leq1$. Show that $$\int_{|z|=1}\overline{f(z)}dz=2\pi i\overline{f'(0)}$$ A hint would be nice. I have tried a few ...
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1answer
27 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
3
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1answer
62 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
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1answer
23 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists ...
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0answers
68 views

What is the limit of this sequence of complex numbers?

Let $z_1$ and $z_2$ be two complex numbers in the upper half-plane. Does the sequence $c_n = \exp^n\left(\sqrt{\log^n(z_1)*\log^n(z_2)}\right)$ converge to a fixed point as $n\to\infty$? If so, what ...
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2answers
40 views

Radius of Convergence for $f(z) = \dfrac{1}{1+z^2+z^4}$ at $\dfrac{1}{2}$

I am practicing some qualifying problems, and I cannot compute the following: Find the radius of convergence $R$ of the Taylor series of $f(z) = \dfrac{1}{1+z^2+z^4}$ centered at $\dfrac{1}{2}$. I'm ...
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2answers
36 views

Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$

Let $a_1, \dots ,a_n$ be arbitrary complex numbers. Define: $f \left( x \right)=\sum_{k=1}^{n}a_ke^{2 \pi ikx}$. I wish to show that there exists an $x\in\left[ 0,1 \right]$ s.t. $f \left( x ...
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71 views

rate of growth of function with specified zeros

One example of this theorem wanted? Let $ \Lambda =\{\lambda_{n}\}_{n=1}^{\infty} $ be a sequence of distinct complex numbers satisfying the following two properties: (1) $ ...
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0answers
84 views

Existence of an entire function with prescribed zeros and asymptotic behavior

One example of this theorem wanted? Let $ \Lambda =\{\lambda_{n}\}_{n=1}^{\infty} $ be a sequence of nonzero complex numbers with the following three properties: There is a constant $ ...
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1answer
67 views

Find 5 odd numbers(1 to 15) to get 30 [duplicate]

---- + ----- +---- + ----- +---- =30 FIll these 5 boxex/dots using 1,3,5,7,9,11,13,15. You can also repeat numbers
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2answers
81 views

An analytic function satisfies $f(1/z)=f(z) $, if $f$ is real on $\{|z|=1\}$, then the coefficients of expansion are real.

An analytic function satisfies $f(1/z)=f(z),\forall z \in \mathbb{C}\backslash\{0\} $, if $f$ has real values on $\{|z|=1\}$, then the coefficients of the Laurent expansion are all real and. Here is ...
5
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1answer
74 views

Completion of Complex Numbers

In some way, $\mathbb{C}$ completes $\mathbb{R}$, why is there nothing that completes $\mathbb{C}$? Is it just more so that we don't want anything more than $\mathbb{C}$, or is there a property of ...
3
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1answer
73 views

asymptotical behavior of integral

I'm interest in the asymptotical of $$\int_{-\pi}^{\pi}\exp\Big((\cos z+i\alpha\sin z-1)t\Big)dz\hspace{3mm}\text{as}\hspace{2mm}t\to\infty$$ for $-1<\alpha<1$. Numberical result suggest that ...
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0answers
31 views

what does projective line of degree one mean?

I know what is projective line, but I'm confused about degree one. Can someone tell me what is "projective line with degree one"?
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3answers
56 views

Show that the complex closed line integral $\oint\frac{\mathrm{d}z}{p(z)}$ is $0$ ($p$ is polynomial)

Let $p$ be a polynomial of degree $n\geq2$ and has $n$ different roots $z_1,\dots,z_n$. Prove that $\oint\frac{\mathrm{d}z}{p(z)}=0$ where the closed path is large enough so that all roots are in the ...
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1answer
63 views

alternate proof of the fundamental theorem of algebra

I was reading over my notes from complex analysis and saw the fundamental theorem of algebra which states that: A polynomial of positive degree over a field $\mathbb{C}$ of complex numbers has a ...
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0answers
133 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
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0answers
29 views

If $f\in H(\mathbb{C})$ and $f(e^\frac{2\pi i}{m}z)=f(z)$, then there exists $g\in H(\mathbb{C})$ such that $f(z)=g(z^m)$.

I have a problem I am trying to solve from an old qualifying exam. It is as follows: Suppose that $w=e^\frac{2\pi i}{m}$ where $m$ is a positive integer, and $f\in H(\mathbb{C})$ satisfies ...
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2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
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0answers
16 views

Conformal map two slits to circles

I am trying to find a conformal mapping that maps a double slitted plane onto a plane with two circles. The two slits are both located along the real axis with similar lengths. For a single slit ...
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5answers
234 views

How to find $\sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)}$

Let $a,b$ be two unequal integers. I have to find the sum below. $$ \sum_{k \in \mathbb{Z}}\frac1{(k+a)(k+b)} $$ I should use complex analysis, but I have no clue where to start. I only now that I can ...
6
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1answer
54 views

Given two holomorphic functions on a region find two other such that…

Let $\Omega$ be a region in the complex plane and let $f_1$ and $f_2$ be holomorphic functions on $\Omega$ having no common zero. Show that there exist holomorphic functions $g_1$ and $g_2$ on ...
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2answers
69 views

Calculating $\int_0^\infty \frac{\ln x}{(x^2+9)^2} dx$

I try to calculate $$ \int_0^\infty \frac{\ln x}{(x^2+9)^2} dx $$ I use a book that tells me to replace $\ln x \ $ by $ \ \ln(|x|) + i\phi_z$ where $\phi_z$ denotes the argument of $z$, chosen between ...
0
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1answer
23 views

There is no holomorphic $f$ such that for all $z$ in a neighborhood of $0$, $f^2(z)=\sin z$

Prove doesn't exist function $f(z)$ which is analytic in neighborhood of $0$ (defined to be $S$) such that $\forall z\in S, f^2(z)=\sin z$. I think that the argument principle (or Liouville ...
1
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1answer
63 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
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3answers
81 views

How to get a function if you have the Fourier coefficients

So I have $$H(e^{i\omega})=\sum_{n=-\infty}^\infty C_ne^{i\omega n}$$ and I know that: $$C_n = \frac{2}{\pi n}\sin^2\left(\frac{\pi n}{2}\right)$$ How can I work out the function that this makes? I ...
1
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1answer
49 views

Why do we define the complex exponential as we do?

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from ...
2
votes
1answer
33 views

More preliminaries of the Martingale Convergence Theorem

Really struggling with this lemma. Not sure about the general structure of the proof. Why have we chosen g to be orthogonal to all functions of the form 4.3.1? Why should $G(\lambda)=0$, does it ...