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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3
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1answer
176 views

Solving the ODE $[(1-x^2)\frac{\partial}{\partial x} - \lambda]f = [\frac{\partial}{\partial x} - \frac{\lambda}{a}]g$

I want to solve $f(x)$ in terms of $g(x)$ in the following ODE $$\left[(1-x^2)\frac{\partial}{\partial x} - \lambda\right]f(x) = \left[\frac{\partial}{\partial x} - \frac{\lambda}{a}\right]g(x),$$ ...
0
votes
1answer
26 views

Symmetric points in $\overline{\mathbb C}$ problem

Statement Let $z_1,z_2,z_ 3$ be three distinct points in $\overline{\mathbb C}$, show that there is a unique line or circle $C$ such that $z_1 \in C$, and $z_2$ and $z_3$ are symmetric with respect ...
0
votes
1answer
54 views

A problem on complex polynomials

Suppose p(z) is a polynomial of degree $n$ having no zeros in $|z|<1$ and $q(z)=z^n \overline{p(1/\overline{z})}$ then, is $|p(z)|<|q(z)|$ in |z|<1 true? May I know why?
2
votes
2answers
2k views

Cauchy distribution characteristic function

I know that it's easy to calculate integral $\displaystyle\int_{-\infty}^{\infty}\frac{e^{itx}}{\pi(1+x^2)}dx$ using residue theorem. Is there any other way to calculate this integral (for someone who ...
0
votes
1answer
37 views

Neighbourhood of a disc

I'm a bit confused on how to write down precisely a neighborhood on an example. My question is the following: Suppose I have a disc $\Omega=\lbrace x\in\mathbb{C}, |x-1|<2.5\rbrace$ and its ...
2
votes
1answer
61 views

Find a conformal mapping from lens to first quadradrant

Consider the disks of radius 1 centered at 0 and 1 in the complex plane. Their intersection forms a lens shape. I want a complex function which is a conformal map from this lens to the first quadrant. ...
7
votes
1answer
336 views

Using normal families to bound a complex integral

I am trying to prove that $$\int_{\partial T(Q)} |F'(z)| \,ds(z) \lesssim \int\int_{T(Q)} |F'(z)| |\varphi'(z)|^2 \log \frac{1}{|z|} \,dx\, dy$$ This is an estimate on page $6$ of this paper by ...
0
votes
1answer
60 views

Laplace's equation periodic in one dimension, from boundary values

I'm trying to solve Laplace's equation in a domain that is semin infinite in one ordinate and periodic in the other. That is, we consider a pair of functions $x(\xi,\nu),y(\xi,\nu)$ such that we ...
33
votes
9answers
4k views

Complex analysis is more “real” than real analysis

In physics, in the past, complex numbers were used only to remember or simplify formulas and computations. But after the birth of quantum physics, they found that a thing as real as "matter" itself ...
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vote
2answers
54 views

Limit of complex numbers' sequence (related to Möbius transformation)

Problem Let $T(z)=\dfrac{7z+15}{-2z-4}$. Let $z_1=1$ and $z_n=T(z_{n-1})$ for $n\geq 2$ Find $\lim_{z_n \to \infty}z_n$ I am having a lot of difficulties trying to solve this. I've tried to find a ...
0
votes
1answer
117 views

How to show that if möbius transformation has an inverse, then it is injective?

Let $f(z)$ be möbius transformation. How to show that if möbius transformation has an inverse, then it is injective? I mean why don't you use this definition to show injectivity of möbius ...
2
votes
2answers
49 views

Find the coefficients $a^{−1}, a_0, a_1$ in the Laurent expansion $\frac{1}{e^z − 1} = ···+a_{−1}z^{−1} +a_0 +a_1z+…$

Find the coefficients $a^{−1}, a_0, a_1$ in the Laurent expansion $\frac{1}{e^z − 1} = ···+a_{−1}z^{−1} +a_0 +a_1z+...$ on $2π < |z| < 4π.$ I know this should be a very easy problem, but not ...
0
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1answer
55 views

How to solve these two differential equation?

I try to solve these two difference equation ; $$ \frac{dq}{dz} = -j\left(b_1q - kp\right),\\ \frac{dp}{dz} = -j\left(b_2p - kq\right) $$ where $j$ stands for $\sqrt{-1}$, and $b_1$ ,$b_2$ and k are ...
0
votes
1answer
80 views

using geometric series rules when solving Laurent Series

when looking at Laurent Series expansions, I sometimes see something like: $$ \frac{1}{\sin z}= ...
0
votes
2answers
309 views

Argument principle and Abel-Plana formula

I find proofs of Abel-Plana formula $\sum_{n=0}^{\infty} f(n)-\int_{0}^{\infty} f(x)\text{d}x=\frac{1}{2}f(0)+\text{i}\int_{0}^{\infty}\frac{f(\text{i}t)-f(-\text{i}t)}{e^{2\pi t}-1}$ where $f$ is a ...
1
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0answers
33 views

Controlling the Sum of a Set of Complex Numbers

Consider a set of N previously fixed angles $\phi_i$. Let $p$ be a positive integer. If $\sum^N_{i=1} e^{ip\phi_i} = 0$, what if any restriction does this place on the value of $p$? If $\phi_i = 2\pi ...
0
votes
1answer
29 views

Asymptotic expansion of $z^{-x}$

Consider the function $z\mapsto z^{-x}$ for $x>1$ (real) and $z$ in the cut complex plane $\mathbb C\backslash\{z\leq 0, \text{ real}\}$. Does this function have an asymptotic expansion of the form ...
1
vote
2answers
63 views

Question about a notation. Norm of the derivative of a function at a point

Given is an analytic function from $M$ to $N$, both equipped with conformal Riemannian metric, say $g$ and $h$ resp. What might the $h$ norm of the derivative of the function at a point mean? ...
2
votes
1answer
96 views

Liouville's theorem and holomorphic function

I'm working on some practice exams and in one I am looking at the following question: Let $f$ be a function holomorphic on $\mathbb{C}$. Suppose that there exist [real] constants $A$ and $B$ ...
0
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3answers
78 views

convergence of an infinite series of complex number

there is the series $\sum\limits_{k=1}^{\infty}\frac{(k^2+i)}{(k+i)^4}$.I wonder how it can be proved of convergence with ratio test
0
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1answer
45 views

Conformal maps on the boundary

Let $\Omega$ be a domain bounded by a closed smooth curve,(i.e $\Gamma=\partial\Omega$), $\mathbb{D}_+=\{z: |z|>1\}$ and $\Omega_+=\mathbb{C}\setminus{\Omega}$. Suppose ...
1
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1answer
68 views

Why doesn't the functional equation imply that $\zeta(s)=0$ for positive even integers?

The Riemann Zeta Function satisfies the functional equation $\zeta(s)=2^s\pi^{s-1}\sin\left(\dfrac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)$. But when $s$ is a positive even number, $\sin\left(\dfrac{\pi ...
3
votes
1answer
355 views

Cross ratio and symmetric points exercise

Problem Let $C$ be a circle or a line belonging to $\overline{\mathbb C}$ and let $z_2,z_3,z_4$. Two points $z$ and $z^*$ are said to be symmetric with respecto to $C$ if ...
0
votes
2answers
23 views

Specify all the singularities of $g(z)$

I am working on some past exam questions, and I just had a question about an intuitive way to know where the singularities of a complex function are. For example for the function ...
0
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2answers
68 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
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vote
0answers
124 views

If $f$ is holomorphic, then there is a holomorphic function $h$ such that $e^{h(z)}=f(z)$

Let $f:G\to\mathbb{C}$ denote a holomorphic function over a star-shaped domain $G$ and $f\ne 0$ on $G$. I want to show that it holds $\frac{f'}{f}$ is holomorphic There is a holomorphic function ...
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vote
0answers
74 views

Uniqueness of holomorphic solutions of a differential equation

Given two polynomials $p,q\in\mathbb C[z]$ consider the initial value problem \begin{align*} f(z)-p(z)f'(z)&=f(z^2)-q(z)f'(z^2),\qquad z\in\mathbb D,\\ f(0)&=0, \\ f'(0)&=1. \end{align*} ...
0
votes
1answer
55 views

convergence radius of taylor series of a complex function in different directions, the same?

Given the taylor expansion of a complex functionf(z) around $z_0$, is the convergence radius of this series the same in different directions, say in real axis ...
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0answers
43 views

Solution to recursion relation using Mellin transform

I have been trying to solve the following recursive equation for $0<x_c<1$ for few a days: $$ P(x) = 2\mathbf{1}_{0\leq x\leq x_c} + 2\int_x^1 dy P\left(\frac{x}{y}\right)y^{-1} ...
1
vote
1answer
38 views

Sequence of Mobius Transformation

Let $ T(z) = \frac {z+2}{2z+1} $. Now it follows that: $ T_1(z) = T(z), T_2(z) = T(T_1(z)), T_3(z)=T(T_2(z)) .... T_{n+1}(z)=T(T_n(z)) $ I'm trying to prove this sequence at the nth terms, but I ...
1
vote
1answer
93 views

For $z \in \mathbb C$, define $f(z) = \frac {e^z}{e^z - 1}$, then

$f$ is entire . the only singularities of $f$ are poles. $f$ has infinitely many poles on the imaginary axis each pole of $f$ is simple. For (1), Since $0$ is a pole of $f$, So $f$ is not entire. ...
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0answers
40 views

Does it have a pole of order $km$ at $z=0$?

Let $f,g$ be meromorphic functions on $\mathbb{C}$. If $f$ has a zero of order $k$ at $z=a$ and $g$ has a pole of order $m$ at $z=0$, then will the order of pole at $z=a$ is $g(f(z))$? ...
0
votes
1answer
57 views

Möbius transformations on $\space \overline{\mathbb R}$

Prove that a Möbius transformation $T(z)=\dfrac{az+b}{cz+d}$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ if and only if it can be written with real coefficients. If it can be written with ...
1
vote
0answers
55 views

Extensions of Weierstrass Factorization Theorem: Essential Singularities and Branch Points

I want to know under what conditions I can take an analytic multi-valued complex function of one variable and rewrite it (while restricting it to $\mathbb{C}$) as a product of a meromorphic function ...
1
vote
1answer
122 views

Radius of convergence of entire function

Let $f$ be an entire function on the complex plane. Is the radius of convergence of $f$ around any point $z_0$ infinite? If so, why? Thank you.
3
votes
2answers
216 views

Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$

Evaluate by contour integration [i am learning complex analysis - calculus of residues] $$\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$$ I tried by taking $x^3$ out from the denominator but that didnt work.
0
votes
1answer
145 views

Find a conformal bijection from $\{z : |z| > 1, Im$ $z < 5\}$ onto an annulus centered at the origin.

I am having trouble doing conformal map problems. Any suggestions on how to do this problem? Thanks. Find a conformal bijection from $\{z : |z| > 1, Im$ $z < 5\}$ onto an annulus centered at ...
2
votes
1answer
62 views

Use contour integration methods to compute $\int_\mathbb{R}\frac{\cos x}{1 + x^2}e^{−ixt}dx$ for all $t > 0.$

Use contour integration methods to compute $$ \int_\mathbb{R} {\cos\left(x\right) \over 1 + x^{2}}\,{\rm e}^{−{\rm i}xt}\,{\rm d}x\,, \qquad \forall\ t > 0 $$ Could someone suggest the proper ...
0
votes
2answers
68 views

Nature of singularities of $f(z)=\frac{1}{\sin(\frac{\pi}{z})}$

What are singularities of $$f(z)=\frac{1}{\sin(\frac{\pi}{z})}$$ I can show that the singularities are given by $$\sin(\frac{1}{z})=0=\sin(n\pi)$$ This gives $z=\frac{1}{n},n=0,1,2,3,4....$ Now, ...
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vote
1answer
92 views

Poles of a function defined in terms of an integral

Suppose $\rho: [0,1] \rightarrow [0,\infty)$ with the following two properties: $$\int_0^1 \rho(x) dx = 1$$ and $$\int_0^1 \rho(x) x dx =\frac{1}{2} $$ Now let $$w(s) \equiv \int_0^1 \rho(x) ...
0
votes
1answer
48 views

Weierstrass m test

Let $\displaystyle f(z)=\sum_{n=0}^\infty a_nz^n$ when the series converges for all $|z|<R$. Let $\displaystyle h(z)=\sum_{n=0}^\infty \frac {a_nz^n}{n!}$. Show that $h$ is an entire function and ...
2
votes
1answer
406 views

When can we switch the limit and the integral?

$\Omega$ is a domain in the complex plane and $F(z,t)$ is a continuous function on $\Omega\times I$ where $I=[0,1]$ is the unit interval in $\mathbb{R}$. Suppose further that $F(z,t)$ is analytic in ...
0
votes
4answers
79 views

complex numbers quadratic equation question

how to solve $z^2 +3|z| = 0 , z$ complex ? treating the complex number as $a+bi $ or anything similar didnt help much...also solving like simple algebric equations also didnt prove effective and ...
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1answer
82 views

Extending a power series of a holomorphic function when the function extends continuously…? [duplicate]

If $f : \mathbb{C} \to \mathbb{C}$ is represented by a power series in $D$ (the unit disc), and $f$ extends continuously into $\bar D$, does the same power series represent $f$ in $\bar D$? I suspect ...
3
votes
1answer
125 views

Möbius transformation: proving the image of the unit circle is a line

Problem 1) Find the Möbius transformation which maps the points $0,i,-i$ to $0,1,\infty$ respectively. 2) Prove that the image of the circle centered at $0$, of radius $1$ is the line $\{Re(z)\}=1$. ...
4
votes
2answers
66 views

Ahlfors Complex Integration

This is my opinion on the question. Is true or not? If not what is the useful solution? Which way is more useful?
2
votes
1answer
52 views

weighted integral in convex hull

Working on an integral $$ J=\frac1{2\pi} \int_0^{2\pi} w(t) g(e^{it}) dt $$ where $\frac1{2\pi} \int_0^{2\pi} w(t) dt=1$ ; $w(t)$ is non-negative continuous ...
1
vote
2answers
71 views

Ahlfors complex integration.

Suppose $f(z)$ is analytic on a closed curve $\gamma$ (i.e $f$ is analytic in a region contains $\gamma$ ). Prove that $\int\limits_{\gamma}\overline{f(z)}f'(z)dz$ is purely imaginary. How can ...
0
votes
2answers
34 views

Maximum value estimation

Let $f$ be an analytic function that is not zero at $\{z:|z|<2\}$. Show that for every natural number $n$: $$\max_{|z|=1}|f(z)-\frac{1}{z^n}|>1$$ I know that ...
1
vote
1answer
48 views

The degree of a map between complex projective lines

Let $P$ and $Q$ be complex polynomials such that $\deg P=p$, $\deg Q=q$ and $\gcd(P,Q)=1$. How can I: show that $F(z)=\frac{P(z)}{Q(z)}$ defines a smooth map $\mathbb{C}P^1\to\mathbb{C}P^1$? ...