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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14 views

Coefficient symmetry of the Laurent expansion of the composition of a function with the Joukowski map

In Sheehan Olver's exposition of how Chebyshev series arise, he lets $f\in C^\infty[-1,1]$ and defines $$g(z)=f(J(z))$$ where $$J(z)=\frac{1}{2}\left(z+\frac{1}{z}\right)$$ is the Joukowsky map. He ...
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19 views

verifying CRE => f(z) differentiable?

Say we had $f(z) = e^{-y}(x\cos x - y \sin x) + i e^{-y}(y \cos x + x \sin x)$, and I verified that the Cauchy Riemann Equations hold, then does that mean I can use the theorem $f'(z) = \dfrac{ ...
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2answers
32 views

CR Equations using Polar Form

I have a question to check whether following function is analytic or not using CR Equations. The question is f(z) = 1/(z-z^5) I just don't know how to start and ...
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43 views

Rotation Number of Polynomial

I conjectured that Maximum Rotation Number of $n$-th degree polynomial image of unit circle (in the complex plane) is $n$. (for example, if $f(z)=z^n$, then rotation number is $n$) Is it right?
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1answer
30 views

Surjectivity of a map $D^{2n} \to \mathbb{CP}^n$

I'm solving an exercise about the complex projective space, and during a step of the solution I'm asked to find a surjective map $D^{2n} \to \mathbb{CP}^n$. I defined the map in this way $$ ...
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27 views

Using Contour Integration to Integrate $\frac {x^{-z}}{1+x}$

I would like to evaluate the following integral using complex integration: $$\int_0^{\infty}\frac{x^{-z}}{(1+x)}dx$$ where $z \notin \mathbb Z$. I'm given that the answer is $\frac{\pi}{\sin(\pi ...
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0answers
18 views

Finding the singular part of a particular rational function

Find the singular part of $$f(z)= \frac{1}{(1+z^3)^2}$$ at $z=-1$. I tried to compute the Laurent series expansion at $z=-1$, as the term with the negative exponent of $(1+z)$ would be the singular ...
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50 views

suggestion on how to learn maths effectively

this question is not about a problem.My problem is I was made to read topics such as real analysis,complex analysis,metric spaces,topology,functional analysis,abstract algebra comprising of group ...
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1answer
42 views

Integration by Residue

How do I evaluate $$\int_{C(0,e)} \frac{1-\cos z}{(e^z-1)\sin z}dz.$$ This looks simple but I have a hard time to find the residue at $z=0$. At $z=0$ the function say $f(z)$ is even undefined. I ...
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+100

Extract real and imaginary parts of $\operatorname{Li}_2\left(i\left(2\pm\sqrt3\right)\right)$

We know that polylogarithms of complex argument sometimes have simple real and imaginary parts, e.g. ...
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1answer
62 views

Gauss curvature of the graphs of the real and imaginary parts of an analytic function. [closed]

General question about analytic function f of complex variable z: $ f(x + i y) = f(z) $. At any mapped point, are Gauss curvatures in separate 3D plots of real and imaginary parts of $ f(x + i y)$ ...
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55 views

complex differentiation

It is my first text here. So I have started to look at complex numbers in death. I do Uni know, so adding $3+4i$ and $4+7i$ is now nothing. What I am stuck on is the idea of taking a derivative of a ...
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1answer
36 views

Questions regarding a complex-analytic function

So the question is formulated as follows. Given the analytic function $z \mapsto f(z) = \dfrac{1}{\sin z} - \dfrac{\cos z}{z}$, Is $z = 0$ a pole, an essential singularity, a removable singularity, ...
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79 views

How To Prove:$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4} = -\frac{7}{{720}}{\pi ^4}$

When I tried to solve this integral: $$\int_0^\infty {\frac{{{x^3}}}{{1 + {e^x}}}} \;{\rm{d}}x$$ I had trouble computing the sieries: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n^4}$$ Thanks.
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1answer
48 views

How we can drow a Blaschke $3$ ellipse?

Today I read the article Ellipses and Finite Blaschke Products (www.jstor.org/stable/3072367 Blaschke ellipses) by Ulrich Daepp, Pamela Gorkin, and Raymond Mortini. In there they have proved very nice ...
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1answer
24 views

Multiple Choice : What will be the values of the function $(e^f)^" (0)$ [closed]

Plz help me to solve this problem. I do not know how to use the property of analytic function to solve the following problem. Let $f$ be analytic function on $\overline D = \{ z \in \Bbb C :|z|\leq ...
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4answers
63 views

First four terms of the power series of $f(z) = \frac{z}{e^z-1}$?

Attempt: $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}$$ $$ e^z - 1 = \sum_{n=0}^\infty \frac{z^n}{n!} -1$$ $$ e^z - 1 = z\sum_{n=0}^\infty \frac{z^n}{(n+1)!} $$ Thus $$ \frac{z}{e^z-1} = ...
3
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1answer
62 views

$f(z)$ is a polynomial $\Longleftrightarrow$ $\lim_{z\to \infty}f(z)=\infty$

Let $f:\mathbb{C}\longrightarrow\mathbb{C}$ be a entire function such that $$\lim_{z\to \infty}f(z)=\infty$$ How to prove that $f(z)$ is a polynomial using maximum modulus principle (without the use ...
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1answer
21 views

Using Cauchy's Theorem on Contour Integral

I need to solve $\int_\gamma (1-e^z)^{-1}$ if $\gamma (t) = 2i + e^{it}$. I would assume Cauchy's Integral theorem applies here, where $\gamma$ is a closed path on a convex open set. I'm having ...
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1answer
19 views

Find a power series of $\frac{\sin z}{z}$ to show it is analytic on $\mathbb{C}$

I know what the power series of $\sin z$ and $\frac{1}{z}$ separately, but I'm not sure how to put them together. Also, I am assuming that when $z=0$ implies $\frac{\sin(z)}{z}=1$. I'm not sure if you ...
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16 views

Complex Analysis- Integral Gamma

Calculate $\int_\gamma (1-e^z)^{-1} dz$ if $\gamma (t)=2i+e^{it}$ Need help getting started.. hints?
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2answers
8 views

Finding the norm of a complex trigonometric function?

Given that the complex norm $|z| = 1$, how would I go about proving that $|cos(z)| \leq e$? Just a hint would be helpful.
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1answer
50 views

Show $f(z) = \frac{z}{e^z-1}$ is analytic in the neighborhood of the origin and find the first 4 terms in its power series representation

In trying to solve the above, I want to use the Cauchy-Riemann equations to show that $f(z)$ is analytic. However, I notice it's not written in the standard form $u(x,y) + iv(x,y)$. How else would one ...
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17 views

Product of the roots of a complex polynomial

[reupload of a previously incorrectly worded problem] Let $p=\sum a_jz^j$ be a complex valued polynomial with integer coefficients of degree $n$, with $|a_n|=|a_0|=1$, and roots $r_1\ldots r_n$. ...
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1answer
32 views

Determining complex function f(z), properties are given … Complex Analysis

I want to determine the complex function $f(z)$ which has the following properties: $\lim\limits_{ z\to\infty} f(z)=3$ $f$ is everywhere analytic except for two singularities: a pole of first ...
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23 views

Information about $f$ from its logarithmic derivative

Let $D$ be the open unit disk in $\mathbb{C}$ and suppose that $f:D\to \mathbb{C}$ is holomorphic. If I know that the logarithmic derivative $\frac{f'}{f}$ can be extended to a meromorphic function on ...
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1answer
27 views

Explanation of how a holomorphic function inside the unit ball can be extended to a meromorphic function in $\mathbb{C}$?

Suppose $f: B(0, 1) \rightarrow \mathbb{C}$ is holomorphic inside the ball, and continuous on the closed ball so that $|f(z)| = 1$ for $|z| = 1$. Prove that $f$ extends to a meromorphic function in ...
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1answer
40 views

Contour integral with cauchy

Calculate integral $$\oint_{\gamma} \frac{e^{2i z}}{z^4}-\frac{z^4}{(z-i)^3}dz$$ when $\gamma$ is circles $S(0,6)$ parameterization once rotated over space $[2\pi]$. Is there more to it than just ...
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1answer
46 views

Why a analytic function must fail to be analytic somewhere on its radius on convergence.

my book wrote this: It says that the function must fail to be analytic somewhere on the circle of convergence. Is this a correct way of proving that this must hold, and is there also a simpler ...
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1answer
52 views

Let $f:\mathbb C\to\mathbb C$ is entire function. And $f(z+1)=f(z)$ and $f(z+i)=f(z)$ then what can you say about $f$? [duplicate]

Let $f:\mathbb C\to\mathbb C$ is entire function. And $f(z+1)=f(z)$ and $f(z+i)=f(z)$ then what can you say about $f$ ? I guessed that it must be a constant because evaluating the function on a ...
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2answers
41 views

Demonstrating that a non-constant holomorphic function attains its maximum on the boundary of its region?

Suppose that $f$ and $g$ are holomorphic in a region containing the disc $|z| \leq 1$. Suppose that $f$ has a simple zero at $z = 0$ and vanishes nowhere else in $|z| \leq 1$. Let $$f_{\epsilon}(z) = ...
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1answer
36 views

Function that is holomorphic on unit disc, bounded, and converges uniformly to 0 given some conditions?

In the proof above what exactly is $\Omega'$ defined as? Is it precisely the part that is NOT within (or on the boundary of) the unit disc? If so, I'm not sure how this proof makes sense -- if ...
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1answer
46 views

Proving that all entire & injective functions take the form $f = ax + b$?

I'm a little confused at both the overall logic in this proof. Are we simply using $g(z)$ to make conclusions about $f(z)$, because $g(z)$ is the reciprocal of $f$? Is the proof assuming that $f$ is ...
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145 views

Regarding the radius of convergence and its equality to a certain limit

Let $f$ be a holomorphic function on the open unit disk $\mathbb{D}$, and suppose that $f$ cannot be extended holomorphically to any open set $\Omega$ containing $\overline{\mathbb{D}}$. Let $f(z) = ...
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2answers
19 views

Determining pole singularity vs. essential singularity?

For the function $$f(z) = \frac{1}{z} + b$$ what type of singularity is $z = 0$? I would say that it is a pole singularity, as $|f(z)| \rightarrow \infty$ as $z \rightarrow z_0$, but a problem that ...
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1answer
102 views

A meromorphic function is open?

I need a hint but please dont tell me that is a generalization for holomorphic functions. Let $G$ open subset connected of $\hat{\mathbb{C}}=\mathbb{C}\cup\lbrace\infty\rbrace$ and ...
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53 views

Derive branch cuts for $\log(\sqrt{1-z^2} + iz)$ as $(-\infty,-1)$ and $(1,\infty)$?

Attempt: First, we examine $\sqrt{1-z^2}$. Note that it can be written $\sqrt{1-z}\sqrt{1+z}$, so the appropriate branch cuts are $(-\infty,-1)$ and $(1,\infty)$ for the inner square root term. ...
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2answers
32 views

Determining the branch of logarithm

I want to determine a branch of logarithm such that $f(z)=L(z^3-2)$ is analytic at $0$. I am not really sure how to find a branch but I will explain few things I tried. Since $z^3-2$ maps $0$ onto ...
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28 views

Trying to find sum of a complex infinite series with Gamma function and factorials

I am trying to find the sum $S$ of the following series. $$S = \sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}t^{2n}\Gamma\left(\frac{1 + 2nH - ...
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32 views

What is the largest open set $\frac{1}{\cos z-2i}$ is analytic in?

This is a very interesting question that I came across and have never solved any question of this sort. How do I find the largest open set on which $\frac{1}{cosz-2i}$ is analytic? Do I find the set ...
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1answer
56 views

What is the Poisson integral formula for?

I have a homework question for complex analysis. Given a harmonic function $u(re^{i\theta})$ on the disk, show that $$u(re^{i\theta}) = \frac 1 {2\pi} \int_0^{2\pi} P_r(\theta - ...
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1answer
27 views

Differential equation $a_0g+a_1g'+a_2g''+\cdots+a_ng^{(n)}=f$ [closed]

Any idea about this problem: If $f:\Omega\subset\mathbb{C}($open simply connected$)\longrightarrow\mathbb{C}$ is a holomorphic function then the differential equation ...
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1answer
10 views

Limits of complex functions

Use the epsilon delta definition to show that $\lim_{z \to 1} \dfrac{i\bar{z}}{2}=\dfrac{i}{2}$. I am not sure how to do this with complex numbers I know I have to do $|z-1|<\delta\implies ...
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2answers
65 views

How far can we take “If $f$ is holomorphic in $D\setminus C$, $f$ is holomorphic in $D$.”?

It is a theorem of Riemann that if a function $f:D\to\Bbb C$ is holomorphic in all but finitely many points where it is continuous, then in fact $\mathcal O(D)\ni f$. An exercise in Remmert's ...
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1answer
31 views

Why uniform convergence on each compact subset implies uniform convergence in the whole open set for analytic functions

I saw there is the argument that given an open region $D$, a sequence of functions $\{f_n\}$ analytic on $D$, and a function $f$ such that $\{f_n\}$ converges to $f$ uniformly on every compact subset ...
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1answer
32 views

Property of analytic functions?

For an open set $\Omega$, a function $f$ is analytic $z_0 \in \Omega$ if there exists a power series $\sum a_n(z-z_0)^n$ centered at $z_0$ with positive radius of convergence, such that: $$f(z) = ...
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1answer
41 views

Integration of $1/|z-z_0|^2$ over a circle in the complex plane

I am trying to prove the following problem $$\frac{1}{2\pi R}\int_{\delta B_R(0)}\frac{|dz|}{|z-z_0|^2}=\frac{1}{|R^2-|z_0|^2|}$$ where $\delta B_R(0)$ is the boundary of a circle with centre (0,0) ...
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39 views

Complex Limit Without L'hopital's

I'm trying to solve for the limit of the following complex function as $z\to0$. I know L'hopital's rule but I'm to find the answer without using that method. The limit is: $$\lim\limits_{z \to 0} ...
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1answer
30 views

Show $\int_0^{\infty} \frac{e^{-x}-e^{-xw}}{x} dx = \ln{w}$ for $\operatorname{Re}({w})>0$

I want to show that for $\operatorname{Re}({w})>0$, $$\int_0^{\infty} \frac{e^{-x}-e^{-xw}}{x} dx = \ln{w}.$$ I've tried setting the problem up as: $$\int_\gamma \frac{e^{-z}}{z} dz = 0,$$ where ...
0
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0answers
31 views

Prove an analytic function is identically $0$

Let $f$ be analytic in a region containing $0$. If $|f(1/n)|<e^{-n}$ for $n>0$, prove that $f$ is identically $0$. My attempt: Write $f(z)=z^kg(z)$. Then $$|f(1/n)|=|(1/n^k)g(1/n)|<e^{-n}$$ ...