1
vote
1answer
23 views

holomorphic functions with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
2
votes
0answers
42 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
1
vote
1answer
33 views

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$

Assume that $f$ is analytic and one-to-one on $\mathbb{D} = \{z : |z| < 1\}$ and $f(z) = z + z^2g(z),$ where $g$ is analytic in $\mathbb{D}.$ Prove that if $f(\mathbb{D})⊂\mathbb{D}$ or ...
2
votes
1answer
53 views

Calculating $\int_0^\pi \sin^2t\;dt$ using the residue theorem

I want to use the residue theorem to calculate $$I:=\int_0^\pi \sin^2t\;dt$$ Since $\sin^2$ is an even function, we've got $$I=\frac{1}{2}\int_0^{2\pi}\sin^2t\;dt$$ The solution of this exercise ...
1
vote
1answer
42 views

an inequality derived from conformal automorphisms of unit disk

Let $f$ be a holomorphic function on $D(0,1)$ such that $|f(z)|<1$ for all $z\in D(0,1)$. I have obtained $$ \frac{|f(0)|-|z|}{1+|f(0)||z|}\leq |f(z)|\leq \frac{|f(0)|+|z|}{1-|f(0)||z|}. $$ Is it ...
1
vote
1answer
32 views

The functions $\{f_n(x) = n\}$ are analytic and each miss the points $-2, -3$. But, they are not a normal family. So what am I missing. Thanks.

Here is a theorem of Montel: Let $\mathcal{F}$ be a family of analytic functions defined on a domain $\Omega$ . If there are two fixed complex numbers $a$ and $b$ that are omitted from the range of ...
1
vote
1answer
52 views

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$. Prove that lim $a_n/a_{n+1} = z_0.$ [duplicate]

Let $f(z) = \sum_{n = 0}^\infty a_nz^n$ be the Taylor series around $0$ of a function which is analytic in $\mathbb{C}$ \ ${z_0}$, $z_0\neq 0$ and has only a simple pole at $z_0.$ Prove that $lim_{n ...
2
votes
1answer
35 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
1
vote
0answers
61 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
0
votes
1answer
35 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
2
votes
2answers
162 views

How to calculate $\int_{-\infty}^\infty\frac{x^2+2x}{x^4+x^2+1}dx$?

I want to calculate the following integral: $$I:=\displaystyle\int_{-\infty}^\infty\underbrace{\frac{x^2+2x}{x^4+x^2+1}}_{=:f(x)}dx$$ Of course, I could try to determine $\int f(x)\;dx$ in terms of ...
3
votes
2answers
72 views

How to calculate $\int_{\partial B_2(0)}\frac{2z^2+7z+11}{z^3+4z^2-z-4}\;dz$?

I want to calculate $$\displaystyle\int_{\partial B_2(0)}\underbrace{\frac{2z^2+7z+11}{z^3+4z^2-z-4}}_{=:f(z)}\;dz\tag{0}$$ Partial fraction decomposition yields ...
3
votes
1answer
62 views

If $f$ has pole of order $m$, then $\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$

Statement: Let $$f(z):=\sum_{k=-\infty}^\infty a_kz^k$$ have a pole of order $m$ at $z_0$. Then $$\text{res}\left(f,z_0\right)=\lim_{z\to z_0}\frac{1}{(m-1)!}\left\{(z-z_0)^mf(z)\right\}^{(m-1)}$$ ...
1
vote
1answer
23 views

A holomorphic function $f$ has an essential singularity in $0$ iff $\exists(z_k)_k$ s.t. $z_k\to 0$ and $|z_k^mf(z_k)|\to\infty$ for all $m$

Let $f:\mathbb{C}\setminus\left\{0\right\}\to\mathbb{C}$ be a holomorphic function $\Rightarrow$ $f$ has an essential singularity in $0$ if and only if $\forall m\in\mathbb{N}:\exists ...
1
vote
1answer
63 views

Prove that A is both open and closed. [closed]

Let $X = \{ z : | z | \leq 1 \} \cup \{ z : | z - 3 | < 1 \} $ be a subset of $\mathbb{C}$. Let the metric be the usual metric $d(x,y) = | x-y |$. Prove that the set A = $\{ z : | z | \leq 1 \}$ ...
1
vote
3answers
81 views

How to get a function if you have the Fourier coefficients

So I have $$H(e^{i\omega})=\sum_{n=-\infty}^\infty C_ne^{i\omega n}$$ and I know that: $$C_n = \frac{2}{\pi n}\sin^2\left(\frac{\pi n}{2}\right)$$ How can I work out the function that this makes? I ...
0
votes
1answer
89 views

If $f(z):=\sum_{n=0}^\infty a_nz^{-n}$ is compact convergent, then $f$ is holomorphic

Let $\left(a_n\right)_{n\in\mathbb{N}}\subset\mathbb{C}$ such that $$f(z):=\sum_{n=0}^\infty a_nz^{-n}$$ is compact convergent on $B_r(0)\setminus\left\{0\right\}$. I want to show: $f$ is ...
2
votes
3answers
68 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
0
votes
1answer
16 views

Cauchy's Integral Formula: conditions vs singularities

I'm sure this is a simple misunderstanding but it was annoying me. So using the version of Cauchy's Integral Formula given on Wikipedia http://en.wikipedia.org/wiki/Cauchy's_integral_formula, it is ...
2
votes
1answer
40 views

Letting $r \rightarrow 1$ in $\frac{1}{2\pi}\int_{0}^{2\pi}\log |f(re^{i\theta})|\, d\theta$

Suppose $f$ is continuous on $\{z: |z| \leq 1\}$, analytic on $\{z: |z| < 1\}$, and $f(0) \neq 0$. For $0 < r < 1$, consider the integral $$\frac{1}{2\pi}\int_{0}^{2\pi}\log ...
2
votes
0answers
61 views

Integrating $xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2})$?

I want to solve any of the two integrals for the complex number $a$ \begin{aligned} I_1 & = \int\limits_{0}^{\infty} xe^{a/x^2 - x^2}\text{Erfi}(x/\sqrt{2}) dx\\ I_2 & = ...
1
vote
1answer
39 views

If $\| \psi \|_2=1$ can I say something about $\| \psi' \|_2$?

If I have a differentiable $L^2$ function $\psi:\mathbb R\rightarrow \mathbb C$ which is normalised $$ \int |\psi(x)|^2\;\text d x = 1 $$ can I say anything about the order of $$ \int ...
1
vote
1answer
36 views

$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z}$

$$\int _{C^{+}(0,3)} \frac {dz}{2-\sin z},$$ $z$ is complex. I have no idea how to solve $2-\sin z$. I will be really grateful for any help
0
votes
0answers
37 views

Is residue may be equal to infinity?

Is residue may be equal to infinity? Is it possible?
3
votes
1answer
114 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
0answers
21 views

curvilinear integral: $\oint_{C^{+}}F dx $ and $F(x_1,x_2,x_3) = [\frac{-x_2}{x_1^2+x_2^2}, \frac{x_1}{x_1^2+x_2^2}, 0]$

$ C=(\cos t \cos\sin(nt), \sin t \cos \sin(nt), \sin \sin(nt)): t \in [0,2\pi]$ a) Find $$\oint_{C^{+}}F dx $$ and $F(x_1,x_2,x_3) = \left[\dfrac{-x_2}{x_1^2+x_2^2}, \dfrac{x_1}{x_1^2+x_2^2}, ...
0
votes
1answer
29 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 ...
0
votes
1answer
54 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 ...
0
votes
1answer
18 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
0answers
56 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 ...
1
vote
1answer
45 views

$\lim\limits_{R \rightarrow \infty}{|\cos z|}$ in complex numbers

Let $z=Re^{i \alpha}$ $\alpha$ is given number and $\alpha \in (0, \frac{\pi}{2})$ Find (if exist) $\lim\limits_{R \rightarrow \infty}{|\cos z|}$ What happen if function $f(z)=|\cos z|$ when ...
1
vote
1answer
25 views

Given a family $\mathcal{F}$ of complex differentiable functions, prove that the derivative at a point is bounded.

We were asked to prove the following claim as part of an alternative proof of the Riemann mapping theorem. Let $G$ be a (proper) simply connected domain, and $z_0\in G$. Consider the family ...
1
vote
1answer
34 views

Laurent series of $\frac{e^{iz}}{z^2+p^2}$, $ p>0$.

I need help finding the main part of the laurent series of $f(z)=\frac{e^{iz}}{z^2+p^2}$ in $ip,-ip$ since these are the two poles of $f$. Due to the orders of the poles are 1 I just have to find ...
1
vote
1answer
30 views

Holomorphic function with Taylor coefficients that tend to 0

Suppose $f$ is holomorphic on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$ and continuous on $\overline{\mathbb{D}}$. If we can write $F(z) = \sum_{n = 0}^{\infty}a_{n}z^{n}$ for $z \in ...
0
votes
1answer
45 views

Operations with $\text{SL}_2(\mathbb{Z})$

We define $\Omega :=\left\lbrace z \in \mathbb{H}\colon -\frac{1}{2} \leq \operatorname{Re}z \leq \frac{1}{2} \wedge |z| \geq 1\right\rbrace$. I want to show that the following holds: $$ \forall \, ...
2
votes
1answer
51 views

If $\gamma$ is a path from $0$ to $1$, what do we know about $\displaystyle\int_\gamma\frac{1}{z\pm i}dz$?

Let $\gamma$ denote a path from $0$ to $1$ which doesn't cross $\pm i$. What can we say about $$\int_\gamma\frac{1}{z\pm i}dz$$
1
vote
0answers
25 views

Linear homogeneous ODE system of first order

Good afternoon. I recently encountered the following problem to which I couldn't find a solution anywhere so far: Given $A:D\to\mathbb C^{2\times 2}$, $D\subset\mathbb C$ open, with holomorphic ...
5
votes
1answer
110 views

Show $f$ is analytic if $f^8$ is analytic

This is from Gamelin's book on Complex Analysis. Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the ...
0
votes
1answer
24 views

the area of the image under a specific holomorphic function of the unit disk

Let $f(z)=z^3+\frac{z^2}{2}$. Let $D$ be the unit disk in $\mathbb{C}$. How to compute $$ Area(f(D))? $$ In the case that $f:D\to \mathbb{C}$ is injective, \begin{align*} Area(f(D))&= \int_D ...
4
votes
1answer
30 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
0
votes
0answers
47 views

A question on particular functions in $L^\infty$

Let D be the open unit disk in the complex plane, $\mu$ be the arc length measure, and $f, g\in L^{\infty}(\partial D,\mu)$ satisfying the following equations: $$ \int_{\partial D} ...
7
votes
3answers
112 views

Integral formulation for the solution of $xy'' + y' = y$

Let's say that $y$ satisfies the following ODE: $$xy'' + y' = y$$ I want to formulate $y$ as a contour integral. I know that the final result I should get is: $$y(x)=\frac{1}{2i\pi} ...
3
votes
0answers
70 views

Series that diverges at infinitely many points on the unit circle

Initial problem Given $A=\{\alpha_1,\dots,\alpha_k\}$ with $|\alpha_i|=1$, does there exist a power series $\sum a_nz^n$ that converges everywhere on the unit circle except when $z\in A?$ ...
5
votes
3answers
333 views

A difficult integral evaluation problem

How do I compute the integration for $a>0$, $$ \int_0^\pi \frac{x\sin x}{1-2a\cos x+a^2}dx? $$ I want to find a complex function and integrate by the residue theorem.
2
votes
2answers
50 views

About Path-connected

Let $a=(a_1,a_2,...,a_k)$ and $b=(b_1,b_2,...,b_k)$ be points in k-dimensional space $\mathbb{R}^k$. A path from $a$ to $b$ is a continuous function on the unit interval $[0,1]$ with values in ...
0
votes
1answer
29 views

Extension of a holomorphic function in the disc

let $f$ be a continuos function in ${0<|z| \leq r} $ holomorphic in the inner and such that $f(z) $ is real for $|z|=r$. Prove that exist a function $g$ on $\mathbb{C} ^*$ such that $f(z) =g(z) $ ...
3
votes
2answers
55 views

Infinitely real-differentiable function with $f(0)=0$ but $\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$

I'm searching for a infinitely real-differentiable function $f:\mathbb{C}\to\mathbb{C}$ with $f(0)=0$ but $$(*)\;\;\;\;\;\int_{\partial B_1(0)}\frac{f(z)}{z}dz\ne0$$ where ...
1
vote
0answers
26 views

$f(z) = u(x,y) + i\cdot v(x,y)$ holomorphic in a connected open set $D$, such that $a\cdot u(x,y)+b\cdot v(x,y)=c$, is constant

Let $f(z) = u(x,y) + i\cdot v(x,y)$ be a holomorphic function in a connected open set $D$. If $a\cdot u(x,y)+b\cdot v(x,y)=c$ in $D$, where $a,b,c$ are real constants which are not all zero, why ...
1
vote
1answer
23 views

A continuos and holomorphic function on $D^2$ that take pure imaginary values on $S^1$ is costant

Let $D := \{ |z| < 1\}$ and $f : \overline{D} \rightarrow \mathbb{C}$ be a continuos and holomorphic function on $D$ that take pure imaginary values on $\partial D$. Why $f$ is constant? From ...
2
votes
0answers
36 views

an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$ \frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k) $$ I simply wish ...