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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39 views

Let $S$ be the disk $|z|<3$ in the complex plane and let $f:S→\mathbb C$ be an analytic function such that $f(1+\frac{\sqrt2}{n}i)=-\frac{2}{n^2}$

Let $S$ be the disk $|z|<3$ in the complex plane and let $f:S→\mathbb C$ be an analytic function such that $f(1+\frac{\sqrt2}{n}i)=-\frac{2}{n^2}$ for all natural numbers $n$.Then what is the value ...
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1answer
37 views

The Convergence of a Complex Power Series

Suppose that $\sum a_nz^n$ has radius of convergence $R$ and let $C$ be the circle $\{z\in\mathbb{C}\mid|z|=R\}$. 'If $\sum a_nz^n$ converges at every point $z$ on $C$, except possibly one, then it ...
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1answer
73 views

About multiplying two essential singularity containing functions

Find analytic function such that f(z) and g(z) both have essential singularity at z = 0 but when multiplied together they have a pole of order 7. I honestly do not recognize essential singularity ...
0
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1answer
24 views

Show that following conditions are equivailent?

$f$ is holomorphic on $\mathbb{C}$. Do the following conditions are equivalent? a) $f(\mathbb{R})\subset\mathbb{R}$ b) $f(\overline{z})=\overline{f(z)}$ Inclusion from b to a: ...
2
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2answers
59 views

finding poles for a complex rational function

So in working out the details of a trig integration with complex integrals problem, I have ended up with an integrand of $$\frac{z}{z^4+6z^2+1}$$ I need to find the roots of $z^4+6z^2+1$ to use the ...
0
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1answer
52 views

Show that $f=\lambda g$ [duplicate]

Let $f$ and $g$ be holomorphic on $\mathbb{C}$ and $|f|\le C|g|$. Show that there exists $\lambda\in\mathbb{C}$ such that $f=\lambda g$. I know I should use Liouville theorem but I don't know how. ...
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0answers
48 views

Two generating meromorphic functions seperate points on a compact Riemann surface?

Problem Suppose $z,f$ are two meromorphic functions on a compact Riemann surface $M$, whose meromorphic function field is $\mathbb C(M)=\mathbb C(z,f)$, where $\mathbb C(M)$ is a finite extension of ...
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0answers
26 views

Proof entire function is constant [duplicate]

$f$ is an entire function, suppose: $$\lim_{z\to\infty}\frac{\Re f}{z} =0$$ Can we proof $f$ is constant? Thanks!
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3answers
34 views

Classification of Singularities of the function

What about the singularities of the function $$\frac{1}{\sin z}$$ I know that at z=$n\pi$, there is singularity, but what about its classification.
1
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1answer
138 views

Why mobius transformation is isomorphic to projective linear group?

I saw on my complex analysis book that linear fractional transformation is isomorphic to the group of invertable 2x2 matrix such that identify scalar multiplication. Verifying that was easy but I ...
0
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2answers
38 views

Question about how to get the residue for infinite amount of poles

So I am asked to find the residue for each pole such as $$ f(z) = \frac{z}{1-\cos(2z)} $$ I understand pole of order 2 with $z= 2\pi k$ excluding zero. I also understand that residue equals to ...
2
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1answer
167 views

How does the Möbius group act on circlines?

This is a continuation of my earlier, rather vague question. I am interested in studying the action of the Möbius group $PGL(2,\mathbb{C})$, on the circlines in the extended complex plane $\mathbb{C} ...
2
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2answers
138 views

Why does this function residue equal 0?

$$ f(z) = \frac{e^{2z}}{(z-1/2)^{2013}} $$ Why does this residue equal 0? If I expand Laurent series, the right side will have $\dfrac{a_{2013}}{(z-1/2)^{2013}}$ $$ + \frac{a_{-2012}}{(z- ...
0
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0answers
463 views

General Question With Examples about Liouville Theorem

Well Liouville's Theorem states that if $f(z)$ is entire function and modulus is bounded then that means that $f$ must be identically constant. I can prove this using cauchy estimate however, my ...
3
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1answer
78 views

Mangoldt Lambda Sum Rearrangement (from proof of Logarithmic Derivative of Riemann zeta function)

Also, we have by the definition of Λ, $$\sum_{n\geq 1} \Lambda(n) n^{−s} = \sum_p(\log p) \sum_{n \geq 1}p^{−ns}$$ (From https://proofwiki.org/wiki/Logarithmic_Derivative_of_Riemann_Zeta_Function) ...
1
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1answer
97 views

Uniqueness of conformal mappings with different normalizations: three boundary points, or an interior point

Some stuff I've seen in lecture but am still a little shaky on: 1) To determine my mapping explicitly, it suffices to know where 3 distinct points on my pre-image object, say, the unit circle, gets ...
1
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1answer
29 views

compute sums of x,y given a condition

Problem: given that $\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=p$, then compute $x+y$ try: i tryed to solve by this way $$\begin{align} ...
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0answers
52 views

Question about integral over $\cos^3(\theta)$ on complex plane

I had an integral of $$\int_{0}^{2\pi}\cos^3(\theta) d\theta$$ The answer came out to be integral over the curve $$\int_{C} \dfrac{(z^2+1)^3}{8iz^4} dz$$ $$=-i* \int_{C}\dfrac{(z^2+1)^3}{8z^4} ...
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1answer
40 views

definition of an conformal map confusion

We said that a conformal map preserves angles, but in lecture we should that the map $f(z) = z^2$ doubles the angle by while mapping line segments to line segments ($z = re^{i\theta}, z^2 = r^2 ...
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1answer
29 views

Classification of the Singularities of the function

The only singularities I can see in the function $$\frac{1}{1+z^3}$$ is the simple pole at $-1,\frac{1+i\sqrt3}{2},\frac{1-i\sqrt3}{2}$. Am I right ? What about the singularities of the function ...
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1answer
47 views

Is $z\mapsto z^{-1}$ conformal?

Check whether the function $$w=\frac{1}{z}$$ is conformal or not, ans discuss how it transforms the points interior and exterior of a unit circle about the origin. For the answer to the first part, ...
1
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1answer
103 views

Schwarz's lemma question.

The next question is from Conway's book first volume page 132: "Suppose $|f(z)|\leq 1 \ for \ |z|<1$ and $f$ is analytic. By considering the function: $g:D \rightarrow D$ defined by: ...
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1answer
41 views

sum over arbitrary subsets

Let $\{a_n\}_{n\in\mathbb{Z}}$ be a sequence of complex numbers. We want to show that $|\sum_{n\in\mathbb{Z}} a_n|$ (or $\sum_{n\in\mathbb{Z}} |a_n|$) is finite. Is it sufficient to show the ...
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0answers
30 views

If $f$ is one to one on $C$ then $f$ is one to one in $C$

Suppose $f$ is analytic inside and on the simple closed curve $C$ and is one to one on the trace of $C$. I need to prove that $f$ is one to one inside the interior of $C$ as well. Since $C$ is ...
2
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1answer
340 views

Using conformal mapping to solve a boundary value problem,

Use conformal mapping to solve the following boundary value problem for $u=u(x,y)$ in the planar region $R=\{(x,y) \in \mathbb{R}^2: x^2 + y^2 > 1 \text{ and } y>0\}$: u solves ...
2
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1answer
158 views

Find a sequence of functions holomorphic on the punctured unit disc that satisfy certain properties about singularities.

Let $(f_n)$ be a sequence of functions that are holomorphic on the punctured unit disc $D'=\{z\in \mathbb{C}: 0<|z|<1\}$, satisfying: (i) For each $n\in\mathbb{N}$, the function $f_n$ has a ...
1
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1answer
92 views

$\sum |f_n|$ converges uniformly implies $\sum f_n$ converges normally

Is the following proposition true? Let $f_n \colon U\subset\mathbb{C}\longrightarrow \mathbb{C}$ be a sequence of continuous (or holomorphic) functions such that: $\sum |f_n|$ converges ...
1
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0answers
50 views

Define $ f:\mathbb C\rightarrow \mathbb C$ [duplicate]

Define $ f:\mathbb C\rightarrow \mathbb C$ by $$f(z)=\begin{cases}0 & \text{if } Re(z)=0\text{ or }Im(z)=0\\z & \text{otherwise}.\end{cases}$$ Then the set of points where $f$ is analytic is: ...
0
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1answer
64 views

Is a constant function considered to be an entire function?

Is a constant function considered to be an entire function? Constant function is differentiable everywhere. Liouville's theorem holds for them too.
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3answers
34 views

Number of Complex satisfying a given condition

Suppose we have a complex number $z$ such that $|z|=1$ and $$|\frac{z}{z'} + \frac{z'}{z}|=1$$ where $z'$ is conjugate . How many complex number satisfy this? So I simplified second condition as ...
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1answer
36 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period ...
0
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1answer
35 views

Basic Geometric Series Question

Calculation of $ \sum_{n=0}^{\infty}2^{2n} z^{2n} $ The answer is We note that the n-th summand has the form $(2z)^n$ Denoting w = 2z The sum is sigma of 0 to n summand being $(w)^n$ which can be ...
0
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1answer
140 views

Finding order of $f(z) = \cos\sqrt z$

What is the order of following entire function: $$f(z) = \cos\sqrt z$$
2
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2answers
77 views

Complex roots forming a equilateral triangle

Suppose we have relation $$z^2 + az + b=0 $$ where $a$ and $b$ are real and roots of this equation $z_1$ and $z_2$ form equilateral triangle with origin then what could be relation between $a$ and ...
0
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1answer
174 views

one-to-one holomorphic map of $\mathbb{C}$ onto itself must be of the form $az+b$? [duplicate]

If $h: \mathbb{C} \rightarrow \mathbb{C}$ is a one-to-one holomorphic map, then $h(z)=az+b$, where $a,b\in\mathbb{C}$. How to prove this argument?
3
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1answer
85 views

Evaluating $\int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} e^{-ixt} \,\mathrm dx$

$$\int_{-\infty}^{\infty} \frac{\cos x}{1+x^2} e^{-ixt} \,\mathrm dx \quad \quad \quad \text{for }t>0$$ Use residue formula, which contour should I try?
5
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1answer
296 views

Residue of $\frac{\cos(\frac{\pi}{z-1})}{z^2 \sin z}$ at $z=1$

Residue of $$\frac{1}{z^2 \sin z}\cos\left(\frac{\pi}{z-1}\right)$$ at $z=1$. More importantly, I don't even know whether it exists or not. The one who creates this question has made questions that ...
3
votes
2answers
132 views

Evaluating $\int_0^\infty \frac{x\sin x}{1+x^2}$ using contour integration?

I'd like to Evaluate $$\int_0^\infty \frac{x\sin x}{1+x^2}$$ The sine function makes the obvious choice $\dfrac{z \sin z}{1+z^2}$ useless since if we integrate over a semicircle sine can become ...
4
votes
1answer
206 views

Mandelbrot and Julia Set

Consider a dynamical system $$z_{n+1}=\frac{\alpha+z_n}{1+z_{n-1}}$$ for $n=0,1,2,\dots$ In other words the system is $$z_{n+1}=f_{\alpha}(z_n,z_{n-1})$$ where $f_{\alpha}$ is defined from ...
1
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1answer
51 views

Prove that $\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$

I need to prove that if $f$ and $g$ are analytic in $D_r(z_0)$ and $g$ has a simple zero at $z_0$, then $$\mathrm{Res}[f/g,z_0]=\frac{f(z_0)}{g'(z_0)}$$ When $f(z_0)\neq 0$ and since $1/g$ has a ...
0
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2answers
70 views

Find cube roots of $-1$

If $-1$ and $\lambda$ are two cube roots of $-1$ find in terms of $\lambda$ the third cube root of $-1$. Am I right in saying that it is just $1-\lambda$?
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2answers
119 views

Polynomial growth, using the Cauchy Integral Formula,

Is this a true statement in Complex Analysis? If a function grows like a polynomial, then it is a polynomial. Or, is it really: if a function grows like a polynomial at infinity, then it is a ...
0
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1answer
26 views

Centers of Argument Principle

https://binyamini.files.wordpress.com/2014/09/last-practice.pdf Q7 Why are there 3 centers A, B, C? I am not sure how there can be 3 centers, I thought the idea was to make the F(Z) plane ...
0
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1answer
42 views

Solving a system of multivariable fifth degree polynomials

Suppose that $$x=u^5 - 10 u^3 v^2 + 5 u v^4$$ and $$y=5 u^4 v - 10 u^2 v^3 + v^5.$$ Given $x,y,u \in \mathbb{R}$, is it possible to find a $v \in \mathbb{R}$ that satisfies the above relations? I ...
3
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1answer
54 views

Explanation of the argument principle.

https://www.youtube.com/watch?v=RRDmCC8gKpY At 22:35 he's trying to explain how $ \arg h(z) $ changes. I am not understanding this though why is $ f(z_0 + Re^{it})$ it not a circle? And also why ...
2
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2answers
86 views

Finding a unique Mobius Transformation

Let $z_1, z_2, z_3$ be three distinct points in $\widetilde{\mathbb{C}}$. (1) show that there is a unique mobius transformation $g$ such that $g(z_1)=0, g(z_2)=1, g(z_3)=\infty$ (2) show ...
3
votes
1answer
151 views

Meromorphic function with a simple pole and a simple zero, and satisfies an inequality. What can it be?

Describe all meromorphic functions f(z) in the complex plane with a simple pole at z=1, a simple zero at z=-1, and for which $$|f(z)|\le M|z|,$$ for $|z|\ge 2$ for some $M>0$. I know that, ...
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1answer
73 views

Can someone please give me some practical application of liouville theorem

All I understand is liouville theorem states if f is entire on the domain specified, and modulus of f is bounded for all z on the domain then f is identically constant. This is all I know and ...
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0answers
102 views

How to convert $ \int_{-\pi}^\pi \frac {d\theta}{(1+\sin^2\theta)} $ to a contour integral?

I want to convert $$ \int_{-\pi}^\pi \frac {d\theta}{(1+\sin^2\theta)} $$ to a contour integral. I know that I can use the substitution $z=\cos\theta + i\sin\theta = e^{i\theta}$ to get $\sin\theta = ...
1
vote
1answer
155 views

Contour integration of the bessel function

The Bessel Function $J_n(x)$ is defined, for a natural number $n$ and real number x, as $J_n(x) = \frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-x\sin\theta)d\theta.$ By using contour integration with ...