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, ...

learn more… | top users | synonyms (2)

0
votes
0answers
9 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
26 views

How to calculate complex integral?

How to evaluate this integral? $$\int_0^\infty\frac{e^{-u t } du}{\gamma^2+[\epsilon(p)-\mu + iu]^2}$$
0
votes
0answers
8 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 ...
0
votes
1answer
15 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 ...
3
votes
0answers
14 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
1
vote
1answer
12 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
10 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
20 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
22 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
16 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
10 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
26 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 ...
3
votes
2answers
41 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 $L^p$ ...
0
votes
0answers
17 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
27 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
24 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
29 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
39 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
40 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
36 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
21 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
36 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
21 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
35 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
358 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
54 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
17 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
23 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 ...
5
votes
0answers
26 views

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. [duplicate]

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. MY ATTEMPT SO FAR: If $f^3$ is holomorphic, then we can ...
1
vote
0answers
11 views

Poisson Integral, when $U$ is discontinuous

So I am working on the following problem. 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 ...
1
vote
1answer
9 views

Writing the Cauchy Integral Formula for functions

In my text it says that: The Cauchy integral forumla for $f(z) = z^2$ yields $$\oint_{|z|=2} \frac{z^2}{z-1}\,dz = 2\pi iz^2 |_{z=1} = 2\pi i$$ Why is $|z| =2$ the path of the integral? Where does ...
2
votes
1answer
29 views

Entire functions satisfying $|f(z)|\geq |\sin z|^{10} $ for all $z\in\mathbb{C}$

How do I find all the entire functions $f(z)$ such that $|f(z)|\geq |\sin z|^{10} $ for all $z\in\mathbb{C}$ Can an entire function have essential singularity? My intuition says that "no". But I am ...
1
vote
1answer
23 views

How to evaluate the integral $\int_C f(z) \, dz$ where $C$ is the unit circle centered at $0$ and $f(z) = \frac{1}{e^z-1}$?

I thought I could use residue theory. I first get the first few terms of the Laurent series as $$ \frac{1}{e^z-1} = \frac{1}{z} - \frac{1}{2} + \frac{z}{12} - \frac{z^3}{720} + \cdots$$ But then I'm ...
0
votes
0answers
14 views

Modulus of parameterized complex function

Say we parameterize the line segment of $0$ to $1 +i$ by $z(t) = t+it$. Now we want to compute $|z^2|$. My text calculates this to be $(\sqrt{2t})^2$ but $$|(t+it)^2|=|t^2+2it^2 -t^2|=|2it^2| = ...
1
vote
1answer
25 views

Intuitively understanding Riemann surfaces

I'm looking at the Riemann surface of $f(z) = z^{1/2}$ so the set $\{(z,w) \in \mathbb{C}^2 : w^2 = z \}$. I understand that the point of the riemann surface is to understand this multi-valued ...
0
votes
0answers
18 views

Please find a Conformal Map [on hold]

Can you find a conformal map from upper half plane onto upper half plane minus double tilted segments(these two segments start from the real axis)? We can find an explicit conformal map from upper ...
2
votes
1answer
28 views

Trigonometric Integrals times exponential

Let's say I want to evaluate the integral: $$\int_0^{\pi/2} e^{ax}\cos^{a}(x) \,dx$$ where $1\leq a \leq 5$ . One standard way to go around would be by applying parts. That would result of course in ...
4
votes
2answers
29 views

Entire function with vanishing derivatives?

Let $f:\mathbb{C}\rightarrow\mathbb{C}$ be an entire function. And assume that at each point, one of it's derivatives vanishes. What can you say about $f$? A hint suggests that $f$ must be a ...
3
votes
1answer
47 views

solving for z in $|e^z| = 2$

How would I solve for z in the following case: $|e^z| = 2$, now I know that $|e^z| = e^a$ if we let $z = a+bi$ so then equating moduli we get $a = \ln{2}$ But what about $b$? $2 = 2e^{(0+2\pi n)i}$ ...
1
vote
1answer
58 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
1
vote
1answer
31 views

How to solve $e^{3z} = 1+i$

I am trying to solve $$e^{3z} = 1+i$$ Putting the RHS into modulus argument form, $$1+i = \sqrt{2} e^{(\pi/4 +2\pi n)i}$$ Now what I want to do is equate moduli and arguments, letting $z = a+bi$ ...
0
votes
0answers
7 views

Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
-4
votes
2answers
29 views

question about a certain inequality implying continuity [on hold]

Let $f$ be a function on $D_1\times D_2\subseteq{\mathbb{C}^2}$, where $D_i=\{z_i:|z_i|<r_i\}$ is such that it satisfies ...