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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9 views

Arbitrary semi-circular path in the complex plane.

I want to try and define a path that starts at $\alpha$ and ends at $\beta$, but gets there by travelling on a circle, anticlockwise. so $\gamma(t) = \frac{\alpha + \beta}{2} + \frac{|\alpha - ...
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1answer
16 views

Discrepancy between text's answer and mine: singular points of $\cot\left(\frac{1}{z}\right) - \frac{1}{z}$

The points $\frac{1}{k\pi}$, where $k \in \mathbb{Z}$ are all singularities of the function $f(z) = \cot\left(\frac{1}{z} \right) - \frac{1}{z}$. My textbook seems to think that they are simple ...
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0answers
15 views

Contour integral along an ellipse

The question reads: Evaluate $$ \int_\gamma f(z)dz, f(z)=2x-3iy: x,y \in \mathbb{R} $$ where $$\gamma(t)=cost+2isint: 0\leq t\leq 2\pi $$ I know that the path formed is an ellipse, but this is the ...
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1answer
60 views

Factoring semiprimes cost estimation

I have two problems that are the following. The first problem is the following: I need to estimate the cost of factorizing a given semiprime based on previous estimations. For example I have the time ...
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0answers
24 views

Find the derivative of complex function $z^{2i}$ at $z=i$ [on hold]

Let $z^\alpha$ represent the principle value of the complex power defined on the domain |z|>0, $-M<\arg(z)<M$ , Find the derivative of complex function $z^{2i}$ at $z=i$.
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1answer
30 views

What is the best way to solve $\lim_{n\to \infty}{(e^{i \theta})^n}$?

What is the best way to solve the limit: $\lim_{n\to \infty}{(e^{i \theta})^n}$ $\theta$ is fixed, but you must have a care for cases $\ \theta > 0 , \ \theta = 0 , \ \theta < 0.$ There ...
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3answers
379 views

Extension of real analytic function to a complex analytic function

I just learned that real analytic functions (by real analytic, I mean functions $f: \mathbb{R} \to \mathbb{R}$ which admit a local Taylor series expansion around any point $p \in \mathbb{R}$) cannot ...
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0answers
17 views

How do I find an upper bound on $z/(z^3+1)$ on a circular path with radius R centred at the origin?

I want to use the estimation theorem, so I want to find an $M$ such that $|\frac{z}{z^3 + 1}| < M$ I cant seem to work with the $z^3$. $$|z^3 + 1| \geq |z^3 - 1| $$ is just not true. How can I ...
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1answer
23 views

Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually ...
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0answers
18 views

principle laurent series of $f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)}$ at $z=0$ and determine $\oint_{|z|=1} f(z)dz$.

Question: So given the function $$f(z)=\frac{(e^z-1)(1-\cos(2z))}{z^4\sin(z)}.$$ First: Give the principal part of the Laurent series of $f$ at $z = 0$. Second: Determine the integral $\oint_{|z|=1} ...
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0answers
25 views

Show that there can only be finitely many numbers $n \in \mathbb{N}$ for which $f(1/n) = 1/(n + 1)$. [duplicate]

Let $f : \mathbb{C} → \mathbb{C}$ be analytic. Show that there can only be finitely many numbers $n \in \mathbb{N}$ for which $$f(1/n) = 1/(n + 1).$$ For this problem I was thinking about using the ...
3
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1answer
54 views

Singular point of $f(z)$ also a singular point of $1/f(z)$ and $f^{2}(z)$

Suppose $z_{0} \in \mathbb{C}$ is an isolated singular point of the function $f$ of a given type (removable, pole of order $N$, essential). I need to show that $z_{0}$ is an isolated singular point of ...
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0answers
18 views

Prove that the point will go 3 times around ellipse

I'd like to prove that if a point $z$ goes once around ellipse with focus $2,-2$ then point $z^3-3z$ goes 3 times around some ellipse with the same focus. I was thinking (since ellipse is a set of ...
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1answer
12 views

Finding the modulus of complex functions

Let $\gamma$ be the path$$\gamma:\left[0,1\right]\rightarrow\mathbb{C}, t\rightarrow\exp\left(t+it\right)$$ I have found that $$\gamma'\left(t\right)=\left(1+i\right)\exp\left(t+it\right)$$ To find ...
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2answers
38 views

$\int^{2 \pi}_0 \frac{1}{3+2 \cos t}dt$ using $\cos t = \frac{1}{2}\left(e^{it} + \frac{1}{e^{it}}\right)$ or using $u=\tan \frac{t}{2}$

Question : Compute the integral of $$\int^{2 \pi}_0 \frac{1}{3+2\cos t}dt$$ I am stucked on this problem since a good while. I think we could convert that real integral into complex integral and ...
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0answers
33 views

Prove the inequality about $Re(z)$

Consider three different vectors $x$,$y$ and $z$ in $\mathbb{C}^{n}$. So $x = (x_{1} \dots x_{n})$ and this is the same for $y$ and $z$. Now we have $\langle x,y\rangle = ...
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1answer
45 views

How do I differentiate an improper integral?

I would like to differentiate a function of the type $\int_x^\infty f(x, t) dt$ with respect to $x$ ($f$ real or complex valued). Does differentiation under the integral sign apply? What are better ...
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0answers
25 views

Necessity of Differential Forms

All the undergraduate and graduate texts on analysis introduce Differential and integral calculus (I will assume this introduction of basic calculus/analysis). Among them, some books also introduce ...
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1answer
22 views

Finding Laurent series

I'm having trouble in finding the Laurent Series of this function: $f(z)=\frac{1-z}{(1-2z)^2}$ Near the point $z=\frac{1}{2}$ I know the answer from Wolfram Alpha, but I don't understand how to get ...
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0answers
31 views

Verifying a Bromwich integral

So I am supposed to verify : $$\frac{1}{1+e^{-x}} = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty} \frac{\pi}{\sin(\pi z)}e^{zx}dz $$ Which is an integral representation of the fermi-dirac equation. ...
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2answers
36 views

Holomorphic homeomorphism of disc

Let $f:U\rightarrow U$ be a homeomorphism of the disc. Then it is not true that f must extend to a continuous map $f:\overline U\rightarrow\overline U$ (there is an example on this site). ($U$ is unit ...
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1answer
23 views

How to determine if an ugly function is harmonic on a domain.

Let $\Omega \subseteq \mathbb{R}^2$ be a domain (open and connected set). By the standard definition, a function $u:\Omega \to \mathbb{R}$ is harmonic $\iff u$ is twice continuously differentiable and ...
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3answers
59 views

$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$, $\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$ - Cauchy integral?

Compute the integrals $$\int^{2 \pi}_0 \frac{1}{ \sqrt{5}+\cos t}dt$$ and $$\int^{2 \pi}_0 \frac{\cos^2t}{ 5-3\cos t}dt$$ I am stucked on these problems since a good while. Is there someone is able ...
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1answer
35 views

Integral and Cauchy theorem

Question : Compute the integral of $$ \int^{2 \pi}_0 \frac{1}{3+2\cos t}dt. $$ Indication: take the path $\gamma: [0,2 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$ and the integral of $$ \int_{\gamma} ...
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1answer
13 views

$\int_{\gamma} \frac{f(z)}{z^3}dz$ - Cauchy formula

Compute the integral $$\int_{\gamma} \frac{f(z)}{z^3}dz,$$ where $f(z)=az^3+bz^2+cz+d$ and $\gamma : [0, 4 \pi] \to \mathbb{C}$, $\gamma(t)=e^{it}$. So by the Cauchy formula $\int_{\gamma} ...
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2answers
36 views

$e^z=3z^5$ - Rouche's theorem

Question : Show that the equation $e^z=3z^5$ possesses five distinct real roots. In using the Rouche's theorem with the function $f(z)=-e^z+3z^5$ and $g(z)=-3z^5$, I succeeded to prove the ...
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0answers
19 views

Isolated Singular Points

I would like to check and see if my reasoning for this question is correct: Find the singular points of the function, and classify them if they are isolated singular points. Also, evaluate if ...
7
votes
1answer
66 views

Why is complex analysis so nice? And how is it connected/motivating for algebraic topology?

This is very much a soft question, but after seeing Cauchy's integral formula in lecture today I was really struck by how neat complex analysis is. I don't understand how all of these amazing analytic ...
2
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1answer
21 views

Is a map that preserves the hyperbolic distance biholomorphic?

Let $\lVert z \rVert_w = \frac{|z|}{1 - |w|^2}$ be the hyperbolic distance in $\mathbb{D}$, and let the hyperbolic metric be $d(z, w) = \inf_\gamma \int_0^1 \lVert \gamma'(t) \rVert_{\gamma(t)} \, ...
2
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2answers
33 views

Can a non-constant holomorphic function take a line to a point?

Title is pretty self explanatory, but given a function $f:D \to \mathbb{C}$, where $D \subset \mathbb{C}$ an open disk containing $0$ upon which $f$ is holomorphic, is it possible for $f(x) \equiv C$ ...
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0answers
21 views

$g(z)=\frac{1}{3-z}$ - Laurent series for two differents annulus

Find the Laurent series for the function $g(z)=\frac{1}{3-z}$ for the annulus $0 < |z|<3$ and $|z|>3$. I understand for the first case, the Laurent series would be $\sum_{n \geq0} ...
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1answer
22 views

$\frac{1}{z}-\frac{1}{\sin z}$ at the origin- Classify singularities

I tried for a while to classify the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked A way to do this it's to consider a hint of a colleague : ...
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1answer
20 views

Classify singularities - Hint [duplicate]

I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?
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1answer
11 views

How do the coefficients in the linear combination of cosines impact the number of local minima of the sum?

Consider the following function: $$f(\theta) = r_0 + r_1 \cos(\theta + \phi_1) + r_2 \cos(2\theta + \phi_2)$$ where $\theta$ is an angle between 0 and $2\pi$. For all $0\leq k\leq 2$ we have $r_k\geq ...
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1answer
41 views

$\frac{1}{z}-\frac{1}{\sin z}$ at the origin - Classify singularities [on hold]

I tried for a while to classifiy the singularities of $\frac{1}{z}-\frac{1}{\sin z}$ at the origin, but I am stucked. Is there someone who is able to help me at this point?
3
votes
1answer
48 views

Prove that $ζ(4)=π^4/90$ knowing that $\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right)$

The question Knowing that: $$\sin(πz) = πz \prod_{n=1}^∞ \left( 1 - \frac{z^2}{n^2} \right) \tag{1}$$ obtain the Taylor series expansion of $\frac{\sin(πz)}{πz}$ to deduce: $$ \sum_{1 ≤ n_1 < n_2 ...
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0answers
25 views

Find and classify singular points of $\cot\left(\frac{1}{z}\right)$

I need to find and classify singular points (i.e., decide whether the point is removable, a pole of order $N$, essential, or not an isolated singular point), including infinity, of ...
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0answers
24 views

What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$?

What is the Laurent series of $\frac{2}{z-1} - z$ in $1<|z|<2$? I can factor the first term to get $$ f(z) = \frac{2}{z} \sum_{n=0}^\infty \frac{1}{z^n} - z $$ where the series converges ...
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1answer
20 views

Holomorphic function with given real part on unit circle

From de Branges' book Hilbert spaces of Entire Functions (page 2): If $h(\theta)$ is a continuous real-valued $2\pi$-periodic function, define $$g(z) := \frac{1}{2\pi} \int_0^{2\pi} \frac{e^{i ...
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1answer
26 views

$f(z)=\frac{1}{z^2-2z+2}$ - Maximum modulus principle

Let the function $f(z)=\frac{1}{z^2-2z+2}$. I have to find $\max_{z \in D(0,1)} |f(z)|$, but I already know that the maxixum would be on $\bar{D}-interior(D)$ by the maximum modulus principle. Is ...
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0answers
30 views

Contour integral of continuous but not holomorphic functions

This question was transferred here following Mathoverflow suggestions. Let us consider two functions $f(z)$ and $g(z)$, both holomorphic on a domain $U$ (a simply connected subset of $\mathbb{C}$). ...
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20 views

$\int_C \frac{dz}{z+1}$ - Specific question

To compute the integral $\int_C \frac{dz}{z+1}$, where $C=C(0;1)$, could it possible to use the Cauchy theorem or I have to compute normally this integral? I know that the integrand is not define at ...
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1answer
14 views

Help in the demonstration of the theorem 1.2 chapter V from Conway's complex analysis book

I'm reading Conway's complex analysis book and I'm stuck in this little detail in the demonstration of this theorem: Why does $|\int _{T_1}g|\le \epsilon$?
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0answers
19 views

Convergence behaviour of Eichler integral

Consiger $g : \mathbb H \to \mathbb C$ a modular form of weight $2-k, k \in \frac{1}{2}\mathbb Z$. Let $z \in \mathbb H$ and consider the following integral: ...
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0answers
18 views

Integration with branch cuts.

This may be a silly question, but when integrating over closed contours in the cut complex plane (a complex plane with a branch cut) do we need to integrate along the branch cut? For instance, if we ...
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0answers
5 views

Proof of decomposition of homotopy into elementary decompositions?

In my Complex Analysis notes, the following lemma is stated without proof: If $G$ is an open connected domain, and $C$ and $C'$ are homotopic in $G$, then the homotopy can be decomposed into a finite ...
6
votes
3answers
84 views

Calculate $\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$ using residues

I'm supposed to calculate $$\int_0^{\infty}\frac{1}{(x+1)(x-2)}dx$$ using residues. The typical procedure on a problem like this would be to integrate a contour going around an upper-half ...
3
votes
2answers
65 views

an analytic function being zero

Let $f$ be an analytic function defined on the unit disc $D=\{z:|z|<1\}$. If $|f(z)|\leq 1-|z|$ for all $z\in D$ then show that $f$ is a zero function on $D$. Please give only hints.
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0answers
10 views

Sum of Roots of Unity With Weighted Exponents

I have the following conjecture that I want to believe has some sort of classical result associated to it, but have yet to find any such evidence. Let $\ell,r\in\mathbb{Z}^+$, and fix ...
0
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0answers
14 views

Uniqueness of a solution to a functional equation

I have two complex-valued functions, $f$ and $g$, that satisfy the following properties. $\overline{x}$ denotes the complex conjugate of $x$ below. $$g(t)\overline{g(t+h)} = f(h) \quad ...