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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2
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0answers
11 views

What are the applications of Dirichelt Beta function and Reimann Zeta function? [on hold]

I always ask myself what are the applications for these two functions.
0
votes
0answers
3 views

Convolution and singularities

I need to find the properties of the solution of a certain functional equation. The equation is of the form $$u\mapsto L[A+F[u]]$$ where $A$ is an entire function, $L$ is a linear operator (which ...
0
votes
0answers
19 views

Why is this true - $\int_C \frac{1}{z^n} dz = 0$ if $n \ne 1$

So I was looking up the reasoning behind the residue theorem and was wondering what was so significant about the $a_{-1}$ coefficient of the Laurent series and I came across this result - $$\int_C ...
1
vote
1answer
5 views

how to show that the set where Real part of an analytic function vanish contains arcs

Consider a function $f$ given by $f(z)=z^3g(z)$, where $g(0)\neq 0$. Then clearly $z=0$ is a zero of function $f$ of multiplicity $3$. Let $A$ be the set given by $\{z: \Re f=0\}$. How do i show that ...
0
votes
0answers
8 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
1
vote
1answer
15 views

Is converse of Lewy theorem true?

In complex analysis, there is a result named Lewy's theorem, which states that: If $u=(u_1,u_2):\subseteq \mathbb{R^2}\to \mathbb{R}^2$ is one-one and harmonic in a neighborhood $U$ of origin ...
1
vote
0answers
5 views

Doubt in a step of the proof of Rado-Kneser-Choquet theorem

I am trying to prove Rado-Kneser-Choquet theorem, which states that if $f$ is sense preserving self homeomorphism of the unit circle $\partial D$. Then harmonic extension $F$ of $f$ is self ...
0
votes
1answer
17 views

Deflating (factoring) a 6th degree polynomial

What is the procedure to factor a 6th degree polynomial of a complex variable? $$P(z)=1+x^2+x^3+x^4+x^5+x^6$$ I do have the correct answer but no idea how to get to it. The answer is: ...
3
votes
0answers
18 views

Are these the correct residues?

$$\int_C \frac{z+1}{z^2-2z} dz$$ for the circle of $\lvert z \rvert = 3 $. Poles are obviously at $ z = {0,2}$. Can I calculate the residues by viewing the fraction in the integral as either $$\int_C ...
2
votes
1answer
14 views

Supremum of the function of a sum for Weierstrass M-test

I have to prove the uniform convergence of $\sum_{k=1}^\infty \frac{k+z}{k^3 + 1}$ on the closed disc $D_1(0)$. Using the M-test, $|\frac{k+z}{k^3 +1}| \leq |\frac{k+1}{k^3 +1}| = \frac{1}{k^2 - k + ...
2
votes
1answer
37 views

Compute an integral..

$\int_\gamma z^n dz$ where $\gamma$ is the unit circle $|z|=1$ oriented counter clockise and $n$ is an integer. Hint: the answer will depend on $n$. What I don't get about it is, if I just apply ...
1
vote
1answer
21 views

derivative of a map $f(A)=AA^*$

Can someone help me with the derivative of this function. I am getting confused $f:GL(n,\mathbb{C}) \rightarrow GL(n,\mathbb{C})$ $A \rightarrow AA^{*}$ where $A^*$ is the conjugate transpose of A. ...
4
votes
1answer
66 views

Please calculate $\sum _{ k=0 }^\infty\left[ \tan^{ -1 }\left( \frac { 1 }{ k^{ 2 }+k+1 } \right) -\ldots \right] $

Not many math problems stump me, but this summation has me stumped. Can someone provide a solution to this summation: $$\sum _{ k=0 }^{ \infty }{ \left[ \tan ^{ -1 }{ \left( \frac { 1 }{ k^{ 2 }+k+1 ...
2
votes
0answers
19 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...
0
votes
1answer
38 views

Conformality of a map

A conformal mapping is a map $f:U\to V$ with $U,V\subseteq\mathbb{C}$ such that the angles are locally preserved. This can be reformulated saying the jacobian matrix is everywhere a scalar multiple of ...
-2
votes
2answers
37 views

Prove that $|\cos(iy)|>\frac{1}{2}e^{|y|}$ [on hold]

We know that for all complex numbers, $\cos^2z + \sin^2z = 1$. This doesn't apply that $|\cos z|<1$ and $|\sin z|<1$. Show that, for all $y \in \mathbb{R}$ $$|\cos(iy)|>\frac{1}{2}e^{|y|}$$ ...
0
votes
1answer
20 views

A sufficient condition for $f$ to have polynomial growth

Let $f(z)=\alpha z\bar z+\beta z+\bar \beta \bar z+\gamma\geq 0, \forall\ z\in\mathbb C$, where $\alpha,\gamma \geq 0$, $\beta\in\mathbb C$. Show that $$f(z)\leq (1+z\bar z)(\alpha+\gamma).$$ I ...
-1
votes
0answers
33 views

How to calculate complex integral? [on hold]

How to evaluate this integral? $$\int_0^\infty\frac{e^{-u t } du}{\gamma^2+[\epsilon(p)-\mu + iu]^2}$$
0
votes
0answers
12 views

Images of some regions of the complex plane by given function?

I'm trying the draw the image of $A=\lbrace z\in \mathbb{C}:-1<Im((1+i)z)<1\rbrace$ by $f(z)=1/z$ and the one of $B=\lbrace z\in\mathbb{C}:|z|<1\rbrace$ by $f(z)=(z-1)^{-1}$. I've managed to ...
1
vote
1answer
30 views

Harnack's inequality

Let $u$ be harmonic on $\{|z|<1+\epsilon\}$ for some $\epsilon>0$ and $u \geq 0$ on $\{|z|=1\}.$ Could anyone advise me how to show $\dfrac{1-|z|}{1+|z|}u(0) \leq u(z) \leq ...
1
vote
1answer
14 views

Contracting closed contours that enclose poles

If $f(z)$ is analytic in a simply connected domain $D$, its integral over any closed contour is $0$. I don't quite understand how the idea of contracting the contour that encloses a pole follows from ...
0
votes
1answer
17 views

Poisson integral with discontinuous $U$

Let $U$ be a piecewise continuous function and bounded for all real numbers. Then define the Poisson Integral for the UHP to be (It can be deduce from the one for the unit circle). ...
0
votes
0answers
4 views

inverse hyperbolic function of a complex argument

It is not too hard to prove that $f(z)=\cosh z$ is a bijection from $$\def\C{{\Bbb C}}D=\{\,z=x+iy\in\C\mid 0<y<\pi\,\}$$ to $$R=\{\,w=u+iv\in\C\mid v\ne0\,\}\cup\{\,w\in\C\mid ...
0
votes
2answers
22 views

Proving complex integral on jordan region boundary equals to zero

Let $D\subset\mathbb{C}$ be a region bounded by jordan curve $\gamma$. Prove that: a. $\int_\gamma z \, dz=0$ b. $\int_\gamma \bar{z} \, dz\neq0$ (hint:$\bar{z}\,dz=(x-iy)(dx+i\,dy)$) ...
0
votes
1answer
24 views

How many roots does a complex polynomial has?

Define $f(z)=z^4-4z^3+8z-2$. Find how many zeros (including multiplicity) the function has in $\{z\in\mathbb{C}:|z|<3\}$. I tried using Rouché's-theorem on $\{z\in\mathbb{C}:|z|<3\}$. The ...
1
vote
1answer
16 views

Analysis of Complex Integration for different $m$

Say we have parameterized and integrated a complex function $z^m$ over a circle of radius $1$ from $0$ to $2\pi$ and get $$\frac{1}{m+1}(e^{i2\pi(m+1)}-1)$$ and say that it is equal to $0$ if $m\neq ...
1
vote
1answer
19 views

Coefficient of an entire funtion under some condition

Let $f(z)=\sum_{j=0}^\infty a_j z^j $ denote an entire function satisfying the estimate$$ |f(z)|\leq M e^{|z|}$$ for all $z\in \mathbb{C}$ for some constant $M$. Prove that the coefficient $a_j$ ...
0
votes
1answer
11 views

Using superposition to reduce a complex solution

This is a solution to under-damped harmonic oscillation: $$x = e^{-(\frac{\beta}{2})t}[cos(\gamma t) \pm i sin(\gamma t)]$$ This is the correct reduction according to wolfram (10) $$ x_1 ...
1
vote
0answers
28 views

Asymptotic expansion of $(\text{log}(1+x))^2$

How can I find asymptotic expansion of the function $(\text{log}(1+z))^2$ with respect to the asymptotic scale $\{z^{-m}, z^{-n}\text{log}(z), z^{-p}\text{log}^2(z), m,n,p=0,1,2,...\}$ while ...
0
votes
2answers
28 views

Are all complex numbers multi-valued/periodic? What about functions?

For example, a complex number like $z=1$ can be written as $z=1+0i=|z|e^{i Arg z}=1e^{0i} = e^{i(0+2\pi k)}$. $f(z) = \cos z$ has period $2\pi$ and $\cosh z$ has period $2\pi i$. Given a complex ...
4
votes
3answers
67 views

How to prove that $L^p [0,1]$ isn't induced by an inner product? for $p\neq 2$

I'd like to know how could i prove that $L^p [0,1]$ isn't induced by an inner product? (For $p\neq 2$, including $p=\inf$). It is clear to me that i would need to find two functions $f$, $g$ in $L^p$ ...
1
vote
0answers
22 views

Sobolev norm inequality.

I would like to prove or to disprove the following statement. Let $u$ and $v$ be functions in $H^{s}(S^1)$, the for every $s'\leq s$ $$\|uv\|_s\leq (\|u\|_{s}\|v\|_{s'}+\|v\|_{s}\|u\|_{s'}).$$ I ...
1
vote
2answers
30 views

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$?

Is $-\log (1-z) = \sum_{n=1}^{\infty}\frac{z^n}{n}$ for $z \in \mathbb{C}, \|z\|=1, z \neq 1$ ? In any case, why?
0
votes
0answers
25 views

Complex Dynamics of the map

Consider a dynamical systems $$ Z_{n+1}=f(Z_n, Z_{n-1}), $$ where $f$ is a mapping from $\mathbf{C}^2$ to $\mathbf{C}$, defined as $f(z,w)=\dfrac{\alpha}{z}+\dfrac{\beta}{w}$. $\alpha$ and $\beta$ ...
0
votes
0answers
30 views

Show that $\sum_{n=1}^{\infty} \frac{z^n}{n}$ converges for $z \in \mathbb{C}$ such that $\|z\|=1$ but $z \neq 1$

I know I could use Dirichlet's test, but I am wondering if the Taylor series of $- \ln (1-z)$ can be used in some way to prove it for $\|z\|=1$, $z \neq 1$. I know the convergence radius is 1 so it is ...
3
votes
2answers
40 views

There is no holomorphic function in $\Omega=\{0<r<\lvert z\rvert <R\}$ with real part $u(x,y)=\frac{1}{2}\log(x^2+y^2)$

Consider $u(x,y)=\dfrac{\text{log}(x^2+y^2)}{2}$ on $\Omega=\{0<r<|z|<R\}.$ Show there is no holomorphic function on $\Omega$ whose real part is $u.$ My attempt: I understand that $u$ ...
5
votes
0answers
41 views

Entire function f which satisfy $|f(z)| \leq |\exp(z)|$ [duplicate]

Can someone confirm whether or not my solution to the following question is okay? Or if I'm missing something Question: Let f be an entire function satisfying: $|f(z)| \leq |\exp(z)| ,\: \forall z ...
5
votes
1answer
40 views

How to prove $\sum_{k=1}^{N} \frac{\sin n\theta}{2^N}=\frac{2^{N+1}\sin \theta + \sin N\theta -2\sin(N+1)\theta}{2^N(5-4\cos \theta)}$

Prove This using De Moivre Theorem $$\sum_{n=1}^{N}\frac{\sin n\theta}{2^n}=\frac{2^{N+1}\sin\theta+\sin N\theta-2\sin(N+1)\theta}{2^N(5-4\cos\theta)}$$ Please help me find my mistake, because ...
0
votes
1answer
22 views

Open mapping theorem in complex analysis - an edge case

Let $f \colon \mathbb{C} \to \mathbb{C}$ holomorphic and not constant. Claim: If $U \subseteq \mathbb{C}$ is open, then $f(U)$ is open. Now by the open mapping theorem, we know that for every ...
1
vote
4answers
37 views

Convergent complex series

Is $$\sum\limits_{n=1}^\infty \frac{i^n}{n} $$ convergent? Im confused as to how to solve this question, I've been trying to use ratio test but that doesn't seem to be helping.
1
vote
1answer
23 views

pole at infinity iff f is a polynomial

I need to show that if f is an entire function it has a polynomial at infinity if and only if it is a polynomial. If I start with a polynomial, it is easy to show that it has a pole at infinity, but ...
1
vote
3answers
22 views

Check whether $f(z)=\Im(z^2)/\bar z$ ($z\ne0$), $f(0)=0$, is analytic or not

Check whether the function defined by $$f(z)=u+iv=\begin{cases} \Im(z^2)/\bar z& \text{if } z\neq 0\\ 0&\text{if} z=0 \end{cases}$$ is analytic or not. My attempt I tried to find the ...
1
vote
1answer
30 views

Is there a non-constant real valued function in $D$ which is analytic in $D$?

Is there exists a non-constant real valued function in $D$ which is analytic in $D$?
2
votes
1answer
11 views

Is mean value theorem for real valued function is hold for complex valued function?

Is mean value theorem for real valued function hold for complex valued function?
0
votes
1answer
16 views

If $f_j\quad (j=1,2,\ldots)$ are analytic in a region $D$ and $\sum|f_j(z)|^2 $ is constant, then can we conclude that $f$ is constant?

If $f_j\quad (j=1,2,\ldots)$ are analytic in a region $D$ and $\sum|f_j(z)|^2 $ is constant, then can we conclude that each $f_j$ is constant?
1
vote
1answer
39 views

Can we conclude that $u^{-1}+iu$ is constant?

If $u$ is a real valued function on $\Delta _R$ and $u^{-1}+iu$ is analytic in $\Delta_R$. Then can we conclude that $u$ is constant?
4
votes
3answers
360 views

How to show that this complex equation has 10 non real roots and how to express them

I did the first part successfully: $$w^{12}=1= \cos 2\pi + i \sin 2\pi$$ $$w= \cos \frac{\pi k}{6} + i \sin \frac{\pi k}{6}$$ Where $k=0,1,2,3,4,5,6,7,8,9,10,11$ I struggled with this ...
1
vote
6answers
56 views

Examples of dense sets in the complex plane

We know that the set $\left\{a+ib:a,b \in \mathbb{Q} \right\}$ is dense in $\mathbb{C}$. Could one give other examples of dense sets in the complex plane?
-3
votes
0answers
20 views

plot the image of the unit circle under the complex mapping $f(z)=iz^3+z-i$ [on hold]

please I need help I need to do this on Matlab: a)plot the image of the unit circle under the complex mapping $f(z)=iz^3+z-i$ b)plot the image of the line segment from 1 to $1+i$ under the complex ...
2
votes
1answer
27 views

Proving entire function is constant

Let $f(z),z^5\bar{f}(z)$ be entire functions on $\mathbb{C}$. Show that $f$ is constant. I tried using Cauchy-Riemann quations in their polar form in order to find out the derivaties are zero and ...