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)

3
votes
3answers
60 views

Bound in Complex Analysis

Can someone direct me towards the right way to approach this problem? Show $$\displaystyle \left|\int_{|z|=R} \frac{Log{z}}{z^2} dz\right| \leq 2\sqrt{2}{\pi}\frac{\log{R}}{R},\; \text{ for } ...
0
votes
0answers
27 views

Confused by a Laplace transform of $f(t)=t^ne^{at}$

Having looked at my lecture notes I was confused by the following part of a derivation of a Laplace transform for the function $\;f(t)=t^ne^{at} ,\quad n\ge0,\; a \in \mathbb{C}, \; f(t)=0 \;\forall ...
0
votes
3answers
48 views

Showing that $\frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right)$

I'm trying to show that $$ \frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy = d\left(\tan^{-1}\left(\frac{x}{y}\right)\right) $$ but am having trouble figuring out exactly how to approach the ...
3
votes
1answer
34 views

How does Ahlfors define derivative on a Riemann Surface?

I'm reading a passage in Ahlfors (3rd Edition page 298) and he says the following: He has previously defined $G_0(f)$ to be the connected component of any germ generated by $f$. Then he wants to ...
3
votes
3answers
70 views

Maximum of an analytic function on the unit disk.

This question is a old question but in that question one condition was not explained well. Let $f$ be analytic on the unit disk $D$. Assume that $f(r)=\max\limits_{|z|=r} |f(z)|$. (Note that here we ...
0
votes
0answers
11 views

If an analytic function has an algebraic order $h$ at infinity then $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity

Assume infinity is not an essential singularity of the analytic function $f$. Then how is $h$, the algebraic order of $f$ such that $\lim_{z\to\infty}z^{-h}f(z)$ is not zero nor is it infinity? p.s. ...
0
votes
1answer
23 views

Isometries of the plane and fixed lines

I am given that for all reflections $g$ there are infinitely many lines $L$ satisfying $g(L) = L$ which makes perfect sense (just take lines perpendicular to the axis of reflection). I am asked to ...
1
vote
0answers
61 views

Are even Dirichlet series constants?

I consider a Dirichlet series with an absolute abscissa of convergence $\sigma$ that can be meromorphically extend to $\mathbb{C}$: $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ and the analytic ...
2
votes
3answers
48 views

Parametrizing curve for complex analysis integral

I'm trying to show that $$ \int_{|z-z_0| = R} (z-z_0)^m \, dz = \begin{cases}0, & m \neq -1 \\ 2\pi i, & m =- 1. \end{cases} $$ Here's my attempt at a solution: We parametrize the curve at ...
2
votes
0answers
35 views

Is this growth condition satisfied by Dirichlet series?

Suppose that we have $a_n=\mathcal{O}(n^k)$ for some $k \in \mathbb{R}$. Thus, the following Dirichlet serie : $$\phi(s)=\sum_{n=1}^{+\infty}{\frac{a_n}{n^s}}$$ is absolutly convergent in the ...
0
votes
1answer
21 views

Images of Regions Under Cayley's Transformation

I'm working on the following problem for my complex analysis course: I can't seem to find Cayley's transformation anywhere in our textbook - could someone clarify to me what it is? I've done a ...
2
votes
4answers
58 views

Showing that Gaussians are eigenfunctions of the Fourier transform

I'm having a bit of trouble on this problem: I've tried to evaluate the integral directly (using the trick from multivariable calculus where you "square" the integral and convert to polar ...
3
votes
1answer
314 views

Find the region where an integral is defined

How to determine where $$f(z)=\int_0^\infty \frac{e^{tz}}{1+t^2} \, dt$$ is defined and holomorphic using Morera's and Fubini's theorem?
7
votes
1answer
136 views

Proving that a function is analytic

I'm struggling with the following problem: Problem: Suppose that $h$ is a continuous function on a simple closed curve $\gamma$. Define $$ H(w) = \oint_{\gamma} \frac{h(z)}{z - w} \, dz. $$ Show ...
1
vote
0answers
12 views

some questions on the soluction of the Dirichlet's problem in the unit disk

Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we ...
0
votes
1answer
34 views

Holomorphic functions and Laplace's equation.

My book says that for any holomorphic function $f(z)=u(x,y)+iv(x,y)$, $u$ and $v$ satisfy Laplace's equation. $f$ is holomorphic $\implies$ * $u_x=v_y$ and $u_y=-v_x$, so ...
0
votes
0answers
33 views

Analytic continuation of a function

Let $$f(z) = A_0 + A_1(z-a) + A_2(z-a)^2 + \cdots$$ converge in some disk $|z - a| < r$. Following Weyl, we magically re-arrange this power series at point $b$ in this disk and the power series ...
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 ...
4
votes
1answer
538 views

Dogbone contour integral/branch cuts/residue at infinity

I am trying to compute: $$\int_0^1 \frac{\sqrt{x-x^2}}{x+2} dx$$ by contour integration. I define $f(z) = \sqrt{z-z^2}$ with a branch cut on $[0,1]$ in such a way that $f(-1)=\sqrt{2}i$, then define ...
3
votes
2answers
636 views

contour integration and branch points

An exercise in a textbook says to evaluate $\displaystyle \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos (ax) \cos^{b} (x) \ d x \ (a > b > -1)$ by letting $\displaystyle f(z) = z^{a-1} ...
0
votes
0answers
88 views

On an application of the Abel-Plana formula

Referring to a previous question, i am having a hard time trying to do the integral: $$f(s)=-i\int_{0}^{\infty}\frac{\log \left[1+\frac{\left(s\log(1+ix) \right )^{2}}{4\pi ^{2}} \right ]-\log ...
0
votes
1answer
45 views

functional equation of entire functions shall have only constant solutions

Given an entire function $f$ with $f'(0)=0$ and a function $g$ holomorphic (at least) in $\mathbb D:=\{z\in\mathbb C\ |\ |z|<1\}$ such that $f*g$ is entire as well and satisfies the functional ...
0
votes
1answer
33 views

Find a conformal map from $\mathbb{D}=\{z;0<\operatorname{arg} z<2π\}$ to $Ω=\{w;0<\operatorname{Im} w<π\}$.

Find a conformal map from $\mathbb{D}=\{z;0<\operatorname{arg} z<2π\}$ to $Ω=\{w;0<\operatorname{Im} w<π\}$. I am having difficulty with this question. Some help would be awesome. ...
1
vote
1answer
61 views

$|f(z)| ≤ 16$ for $|z| = 4.$ Prove that $|f(3i)| ≤ 9.$

(a) Suppose that $f(z)$ is analytic for $1 ≤ |z| ≤ 4.$ Assume that $|f(z)| ≤ 1$ for $|z| = 1$ and $|f(z)| ≤ 16$ for $|z| = 4.$ Prove that $|f(3i)| ≤ 9.$ (b) Prove that there is no non-constant ...
0
votes
2answers
50 views

$n$-to-$1$ near zero of holomorphic function

Can someone explain to me why a holomorphic function that grows like a polynomial of degree $n$ is $n$-to-$1$ near it's roots? I keep reading this fact on this site, but I can't find an explanation.
1
vote
0answers
42 views

Show that $f(z)= \sum_{n=0}^{\infty} z^n$ is analytic in $|z| <R$ [closed]

Let $f(z)= \sum_{n=0}^{\infty} z^n$ with $|z| <R$ where $R$ is the radius of convergence of $f$. Then show that $f$ is analytic in $|z|<R$.
0
votes
2answers
26 views

Cauchy integral formula or something else?

I need to determine the function $\;f(z)$ if $$f''(z)=\oint_{\partial C_1(0)}{\sin^2\xi \over\left(\xi-z\right)^3}\mathbb{d}\xi$$ with $C_1(0):\left|z\right|<1$ positive. Additionally ...
1
vote
2answers
36 views

Show that $\sum_{k=1}^{n}a_ke^{2 \pi ikx}$ has a root in $\left[ 0,1 \right]$

Let $a_1, \dots ,a_n$ be arbitrary complex numbers. Define: $f \left( x \right)=\sum_{k=1}^{n}a_ke^{2 \pi ikx}$. I wish to show that there exists an $x\in\left[ 0,1 \right]$ s.t. $f \left( x ...
3
votes
2answers
58 views

If $f'(z_0)\neq 0$ then $f$ has an holomorphic inverse.

Problem: Let $U\subset\mathbb{C}$ be an open set, $f:U\to\mathbb{C}$ an holomorphic function of class $C^1$ and $z_0\in U$. Prove that if $f'(z_0)\neq 0$ then there exists a neighborhood $V$ of $z_0$ ...
1
vote
0answers
21 views

Properties of a specific Complex function

Consider a map $f_{p,q}$ from $\mathbb{C}^2$ to $\mathbb{C}$ is defined as $f_{p.q}(z,w)=\frac{p+q.z}{1+w}$ where the $p$ and $q$ are two complex numbers. What can we talk about continuity, ...
3
votes
0answers
52 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
6
votes
1answer
34 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
0
votes
0answers
24 views

Biholomorphic, Hypersurface

I'm learning the Hypersurface. And my teacher has a question: Find an example such that two Hypersurfaces are biholomorphic. I think that $$A=\{(x,y)\in \Bbb C,\ \rho(x,y)= x^2+y^2-1=0\}$$ and ...
1
vote
1answer
34 views

If $\zeta$ is a function of characters what does it mean for it to be regular?

This is from lemma 2.4.1 of Tate's thesis. Lemma 2.4.1: A $\zeta$-function is regular in the "domain" of all quasi-characters of exponent greater than $0$. proof: We must show that for each ...
1
vote
3answers
48 views

integrating $\int_{\gamma}e^zdz$ with $\gamma$ is the arc on the unit circle that unites one with i

I am stuck integrating $$\int_{\gamma}e^zdz$$ with $\gamma$ is the arc on the unit circle that unites one with i. I tried this : The integrand $\mathrm{e}^z$ is holomorphic for $\vert z \vert \le ...
2
votes
2answers
38 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then find whether the above series ...
0
votes
1answer
16 views

Let $A = \{1/2 < |z| < 2\}.$ Is there an analytic function $f$ on $\mathbb{C} \setminus \{0\}$ so that $Im(f) < −1$ on $∂A$ and $f(1) = 0$?

Let $A = \{1/2 < |z| < 2\}.$ Is there an analytic function $f$ on $\mathbb{C} \setminus \{0\}$ so that the imaginary part $Im(f) < −1$ on $∂A$ and $f(1) = 0$? Explain your answer. I am ...
3
votes
1answer
30 views

conformal mapping onto right half plane

Find a conformal map of $D:=\{z\in\mathbb{C}:|z-i|<\sqrt{2}$ and $|z+i|<\sqrt{2}\}$ onto the right half plane. My idea was to use $$f(z)=\frac{z+\sqrt{\sqrt{2}-1}}{z-\sqrt{\sqrt{2}-1}}$$ To ...
0
votes
0answers
17 views

Writing a Mobius transformation as two fonctions belonging to a specific set

I had to prove something about the following set of maps: $$ H \quad = \quad \{ z \ \mapsto \ \frac{\rho^2}{\bar{z}-m} + m \ : \ m, \rho \in \mathbb{R} \} \quad \cup \quad \{z \mapsto -\bar{z} +2 ...
5
votes
1answer
48 views

Composition of an analytic function with a continuous function that is analytic

If $f$ is a continuous function such that $g(z)=\sin{f(z)}$ is analytic, then is $f$ analytic? I know we can take $f(z)=\bar{z}$ then $f$ is continuous but $g$ is not analytic. Same holds if we take ...
1
vote
1answer
29 views

Order of a zero of a complex polynomial

Is there a quick and easy way to determine an order of a zero $z_0$ of a complex polynomial without having to derive it $n$ times and check if $\;f^{(n)}(z_0)=0$ or not, which requires a lot of ...
0
votes
0answers
42 views

Finding a homeomorphism between these two balls

Let $u_1,u_2,u_3 \in \Bbb C$ be the cubic roots of unity. Define two norms on $\mathbb{C}^2$, $$\Vert (x,y) \Vert_1 = \sqrt{\vert x \vert^2 +\vert y \vert^2} \ \text{and} \ \Vert (x,y) \Vert_2 = ...
7
votes
6answers
400 views

How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
10
votes
2answers
374 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
1
vote
1answer
56 views

Rouches Theorem Applied to a family of Polynomials

I would like to prove that the family of polynomials $z^{2j+2} + \alpha z^{2j+1} - \alpha z - 1$ has only one root inside the open unit circle when $|\alpha|$ is greater than 1. This seems like an ...
1
vote
1answer
26 views

On Stein manifolds and constant functions

Stein manifolds are defined here: http://en.wikipedia.org/wiki/Stein_manifold#Definition Obviously, M is Stein implies that there is a non-constant holomorphic function defined in it. Is the converse ...
10
votes
6answers
11k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
0
votes
4answers
91 views

Shown that $f(z)=\left | z^2-4 \right |^2$ is holomorphic

Shown that $f(z)=\left | z^2-4 \right |^2$ is holomorphic. I need to prove that $f(z)$ is holomorphic or not. Well first I need to convert $f(z)$ in terms of $x,y$ but I dont understand how to do it. ...
2
votes
3answers
113 views

Integration by Euler's formula

How do you integrate the following by using Euler's formula, without using integration by parts? $$I=\displaystyle\int \dfrac{3+4\cos {\theta}}{(3\cos {\theta}+4)^2}$$ I did integrate it by parts, by ...
1
vote
1answer
37 views

Confused by the task given (involves identical inequality of functions)

The task says: Show that if some function $\;f(z)={1\over g(z)}$, where $g\not\equiv0$ is an entirely analytic function, then the isolated singularities of $\;f$ are exactly zeros of $g$ ...