Tagged Questions

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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Finding a Mobius transformation

Let $R=\{z\in \mathbb{C}: Re(z)>0, |z-3|>1\}$ and $A=\{z \in \mathbb{C}: 1<|z|<p\}.$ Find a Mobius transformation $f$ and $p$ such that $f$ maps $R$ conformally to $A.$ May I verify if my ...
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27 views

Equations with modulus of a complex variable

I am struggling a bit to solve equations involving the modulus of complex variables. I am given the equation $|z-z_0|=|1-z_0z^*|$, where $z$ is a complex variable, $z_0$ is a complex number and $z^*$ ...
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1answer
32 views

Question about laurent series

Find laurent series for $f(z)=\dfrac{1}{z^2-1}+\dfrac{1}{z(z-1)};z_0=0$ that converges in $0<|z|<1$. I tried to find the solution for the first fraction like this. \begin{align} ...
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1answer
49 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.
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1answer
16 views

Help needed to establish a conformal mapping

Could anyone advise me on how to find a conformal map from $H=\{z \in \mathbb{C}: Re(z)>0\}$ to $A= \{z \in \mathbb{C}:|z|>1, |z-2|<3\} \ ?$ I tried to compose the map in terms of ...
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0answers
22 views

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$?

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$ ?. I know that zero is an essential singularity, and so $f(D(0,\delta)-\{ 0 \})$ is dense in $\mathbb{C}$. This question ...
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1answer
20 views

Laurent series at z = 0

I want to determine the Laurent series around z = 0 (so a Maclaurin series I think) of the following function $f(z) = 4/(z^2+2z-3)$ which converges in $z = 1 + i$ I can rewrite the function as ...
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1answer
33 views

determining a residue by taking a limit

To determine a residue, I need to take this limit: $$\lim_{z\to 2\pi ik} \frac{d}{dz}\frac{(z-2\pi ik)^2}{z(e^z-1)^2}$$ with $k$ any integer number (like -1, -1, 0, 3, 7) I have tried l'Hopital's rule ...
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1answer
40 views

Complex integrations

Calculate integrals $$\oint_{|z|=1} \frac{z^2 e^z}{2z+i} dz $$ and $$\oint_{|z|=2} \frac{e^z}{z^2+z} dz $$ These are simple integrals to do with cauchy integral theorem right? First one. ...
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1answer
48 views

Proving that an analytic function is $0$

We are given an analytic function $f(z)$ in the region $\Omega=\{z : b>Re(z)>a\}$. It is also given that the function is continuous and bounded in $\overline\Omega$. The question is to show that ...
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1answer
22 views

Möbius transformation $(-1,0,1)\to (i,-1,-i)$

Find Möbius transformation $S$ that maps points $(-1,0,1)$ to points $(i,-1,-i)$. And what's the image of real axis and upper half of imaginary axis $\{z\in\mathbb{C}| \operatorname{Im} z \geq 0 \} $ ...
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1answer
26 views

why $G(z)=\frac{f(z)}z, z\ne 0; f'(0), z=0$ is holomorphic?

I'm reading Gong Sheng's Concise Complex Analysis to get some basic understanding. On $\S 2.5$ page 66 Theorem 2.19 (Schwarz Theorem) it says If a holomorphic function $f(z)$ maps the unit disc ...
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1answer
17 views

holomorphic function power series: $f(z)=\sum_{j=0}^\infty a_j(z-z_0)^j$, can it be extended to $U$?

It's well know that if $f(z)$ is holomorphic in $U\subseteq \mathscr C$, then $f(z)$ could be expanded as a power series $$f(z)=\sum_{j=0}^\infty a_j(z-z_0)^j$$ in $D(z_0,r)$, if $z_0 \in U$, $\bar D ...
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1answer
31 views

Question about quadratic equation of complex coefficients.

Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$ If it is given that $|z_1|\not=|z_2|,$ how can I obtain the condition for this containing $a,b,c?$ ...
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12 views

How to find magnitude of complex fractions function

Could you help me to find square of magnitude of complex fraction function that given by $$G=\frac {s+2}{s^2+2s+2}$$ where $s=j\omega$ Thank all This is my solution $$|G|^2=\frac ...
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66 views

Why is such a the series algebraic but rational?

The coefficients of the series expansion of the algebraic function $A=\frac{1-\sqrt{1-8x^2}}{4x}$ are all intergers: $$A(x)=x+2x^3+8x^5+\cdots$$ But according to Polya's research,if $ F(x)$ is a ...
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1answer
41 views

Is it an open problem about Riemman Hypothesis non-trivial zero? [duplicate]

Let's assume RH was correct, and $1/2+Ki$ is any one of non-trivial zero of $\zeta$, is following problem open? 1) $K$ is irrational number 2) $K$ is transcendental number
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0answers
22 views

contour intrgration, what's the right answer?

There exists an integral as follow: $$ \bar G(t)=\int_{-\infty}^{\infty}\frac{dE}{2\pi\hbar}e^{-iEt/\hbar}\frac{1}{E-\epsilon+i0^{+}} $$ My solution is: $$ {2\pi\hbar}\bar G(t)=-i\pi e^{-i\epsilon ...
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0answers
27 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
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2answers
28 views

Algebraic expressions with complex coefficients $(1-i)z^2-2iz-4=0$

How does one solve expressions such as $(1-i)z^2-2iz-4=0$ Own attempt $$\begin{align} &z^2-\frac{2iz}{1-i}-\frac{4}{1-i}=z^2-z(1-i)-(2+2i)=0\\\iff&z^2-z(1-i)-\frac{i}{2} = 2+\frac{3i}{2} ...
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0answers
11 views

Link between complex conjugate and exeptional complex differentiability

Complex differentiability has remarkable differences to real differentiability and came to hear that the reason is: With the complex conjugate, there exists a non trivial field automorphism that is ...
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3answers
44 views

Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$

Find the Laurent series of $f(z) = \frac{1}{z-2} + \frac{1}{z-3}$ for $2 < |z| < 3$ and for $|z| > 3$. Is the first step here to notice that $$ \frac{1}{z-2} + \frac{1}{z-3} = ...
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1answer
22 views

Dirichlet character modulo p

How can I prove that if $\chi$ is a non-principal character modulo $p$ prime, then $\chi (-1) = \overline{\chi} (-1)= \pm 1$ and $\sum_{x=1}^p \chi (x) e^{2\pi i x}=0$? For the first question, I just ...
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2answers
33 views

Continuously extended holomorphic function on the unit disc.

Let $f$ be continuous on $\bar{\mathbb{D}}$ and holomoprhic on $\mathbb{D}$. How can we show that $$\int_{\partial \mathbb{D}}f(z)dz=0$$?
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2answers
19 views

Question about convergence in complex numbers field

It may be a simple question, but if we want to show that $(z_n)\subset\mathbb{C}$ is convergent to $z\in\mathbb{C}$ then we should just check that absolute value of $z_n$ is convergent to absolute ...
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1answer
35 views

Symmetry in complex plane

In a book I am reading, symmetry about a curve in complex plane is defined as follows: Let $F(x,y)=0$ be a simple curve. Then points $z, z_0$ are symmetric about this curve iff $ F \left( ...
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1answer
37 views

Convergence of $\sum_{n=1}^{\infty} \frac{1}{n^z}$

Let us consider $z\in \mathbb C$; what is the condition on modulus of z in order that $$\sum_{n=1}^{\infty} \frac{1}{n^z}$$ the series (zeta function?) converges? For example, if $|z|=1$, the series ...
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1answer
37 views

My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
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0answers
32 views

A problem on analytic function [duplicate]

Let $f(z)$ be analytic on $D=\{z\in\mathbb{C}:|z-1|<1\}$ such that $f(1)=1$. If $f(z)=f(z^2)$ for all $z\in D$, then which one of the following statements is not correct? (i) $f(z)=[f(z)]^2$ for ...
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0answers
33 views

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]

Calculating the power series expansion about pi/2 of g(z)=tan[z/2]. Now calculate the expansion about 0. I'm having trouble doing this. I'm not even sure which is the best way to approach it, for ...
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1answer
30 views

Uniform convergence of the power series except at the point 1.

I couldn't solve the following problem from Lieb's Complex Analysis. Let $a_k$ be a decreasing sequence of real numbers that converge to $0$ and suppose that the radius of convergence of the series ...
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1answer
24 views

Orthogonal parameterization

Consider the function $$f(a,b,c,d):=\frac{\left(a^*\right)^2b^2-\left(b^*\right)^2a^2+\left(c^*\right)^2d^2-\left(d^*\right)^2c^2}{a^*a+c^*c}$$ With complex parameters $a,b,c$ and $d$ Now find any ...
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0answers
48 views

Can this be expressed by a contour integral? [on hold]

Let $f(z)$ be a real entire function of the form $f(z) = a_1 z + a_2 z^2 + ...$ such that $0 < a_{n+1} < a_n$. Consider $g(x) = f^{-1}(f(x)-f(x-1))$ where $x$ is a positive real and $f^{-1}$ ...
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1answer
36 views

Calculate integral of $\ln(z)$ using the residue theorem

Please is it possible to calculate $\int_{C(0,1)}\ln(z)\,dz$ using the residue theorem? Thank you for your help.
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0answers
25 views

What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
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0answers
78 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
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1answer
37 views

A holomorphic function with non-vanishing derivative

I really want to understadn the proof of the following theorem from Lieb's Complex Analysis: Let $f:U\rightarrow \mathbb{C} $ be a holomorphic function with non-vanishing derivative. Then: For ...
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3answers
32 views

Prove that $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic

Suppose that $u,v$ are harmonic functions on doman $D$, and they are harmonic conjugate. Prove that function $\phi(x,y)=e^{u(x,y)}cos(v(x,y))$ is harmonic on $D$. What I've done was to take the ...
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0answers
36 views

Prove $U$ is subharmonic?

My attempt: Integration by parts says $\int u \triangle \varphi=\int\triangle u \varphi$. We know the left hand side is always $\ge 0$, and hence $\int\triangle u \varphi \ge 0$, since $\varphi \ge ...
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0answers
20 views

Determine f(z) by evaluating the sum

Determine an explicit expression for $f(z)$ by determining the sum of the series $f(z) = \sum_{n = 1}^\infty \frac{1}{n}$ $\cdot (\frac{z}{z-1})^n$ where $z\ne 1$ Yeah... I really don't know where ...
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0answers
24 views

conformal map of a portion of unit disk onto upper half plane

How do we construct a conformal map from $\{z=x+iy,x>1/2,|x+iy|<1\}$ onto the upper half plane? My idea is first create a sector sending one of the two intersection points to infinity.Any help ...
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2answers
44 views

$\tan(z)$ with residue theorem

Calculate $$\oint_{|z|=2}\tan(z)\,dz$$ because $\tan(z)=\dfrac{\sin(z)}{\cos(z)}$ the poles are when $\cos(z)=0$ at $z=\pm\pi/2\pm n\pi, \;n\in\mathbb{Z}$ Poles inside $|z|=2$ are $\pm\pi/2$ and ...
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1answer
24 views

$\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$\{0}

I need to prove the set identity of the complex logarithm $\log(z_1z_2)=\log(z_1)+\log(z_2)$ where $z_1,z_2\in \mathbb{C}$. ...
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1answer
20 views

Function vanishes identically on the unit disc

Let $f$ be analytic in $\mathbb D$ and continuous in $\overline {\mathbb D}$ . Let $A$ $=$ {$ e^{i \theta}| |\theta - \pi |< \epsilon$} , for $\epsilon > 0$ small enough such that $f|_{A} = 0$ . ...
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0answers
26 views

Covergence to 0, while imaginary part is bounded

Given $f$ to be bounded and holomorphic on $\{z \in \mathbb C \mid -\pi < \operatorname{Im} (z) < \pi\}$. Let $\lim \limits_{x \to \infty} f(x) = 0$, where $x \in \mathbb R$ . Then prove that : ...
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1answer
19 views

Convergence in upper half plane.

Consider the upper half-plane $\mathbb H^{+}$ & let $f$ be a bounded holomorphic function on $\mathbb H^{+}$ . If $lim _{t \to \infty} f(it) = 0$ ; prove that: $lim _{t \to \infty} f(tz) = 0$ ...
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1answer
14 views

the crossover point of four complex points

If there is four complex points $z_1,z_2,z_3,z_4$ in complex plane $\mathbb{C}$, I want to get the crossover point of the line $z_1z_2$ and $z_3z_4$. If I use the $Re(z_i)$ and $Im(z_i)$, it is easy ...
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0answers
21 views

Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant. [duplicate]

Let $f$ be entire in $\Bbb C$. If $Re(z)>0$ $\forall z \in \Bbb C$. Then prove that $f(z)$ is constant. Please read below before marking duplicate. If $Re(f(z))>M$ then taking $|\frac ...
1
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1answer
26 views

If w is a complex root of 1. Find the value of w^4+w^8

If $w$ is a complex root of 1. Find the value of $w^4+w^8$ Why does complex root of 1 always mean that $w^3=1$ ? Why not $w^2$ ? Back to the question, here's what I did: ...
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2answers
24 views

Integral with residues

Calculate integral $$\oint\limits_{\gamma}\frac{e^z}{z^4+5z^3}dz$$ Where $\gamma$ is parameterization of one rotation of circle $A(0,2)$ So if I write the integral like this ...