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
3answers
45 views

An analytic function on the complex plane minus a discrete set of points

I'm stuck on the following question: Let $A$ be a discrete set of points in $\mathbb{C}$, and $f$ analytic on $\mathbb{C} \setminus A$ such that $f$ has a simple pole at each $a \in A$, and the ...
3
votes
2answers
50 views

Is there a simpler way to compute the residue of a function at a pole of order 3?

The function $$\frac {1}{z^2(e^{i2\pi z}-1)}$$ has a triple pole at z = 0. To compute the residue of f at z = 0, I can compute the Laurent expansion of f about z = 0, and then read off the ...
2
votes
1answer
42 views

Do conformal mappings other than the Mobius transformations preserve symmetry?

Linear fractional transformations preserve symmetric points, e.g., if the real axis in the z-plane gets mapped to the imaginary axis in the w-plane, then points symmetric with respect to the real axis ...
1
vote
0answers
33 views

Why does the upper half plane get mapped inside of the polygon?

If a conformal mapping, e.g., a Schwarz-Christoffel mapping, maps the real line (from left to right) to a polygon, which is traced out from left to right, why is the upper half plane mapped to the ...
1
vote
0answers
25 views

Conformal mapping from upper half-plane to unit disk

I have run into some confusion while reading Newman Bak's Complex Analysis. The text states that if we wish to determine a comformal mapping $h$ of the upper half-plane onto the unit disk, assuming ...
1
vote
1answer
29 views

Example of polynomial in dynamics

I am looking for an example of a post-critically finite polynomial $P$ (i.e. all critical points have finite orbit), which has both the following: a critical point on its Julia set (such as the ...
2
votes
1answer
35 views

Strong maximum modulus principle

(Weak) maximum modulus principle states that if $f$ is a non-constant holomorphic function on some domain $D$ in $\mathbb{C}$ then $|f|$ can't have global maximum in $D$. Strong version says that ...
1
vote
1answer
34 views

Question about Entire functions

Let $D=B(z_0,R)$ be the open disc centered at $z_0$ with radius $R>0$ and $f$ be a non-constant entire function. Is it true that $f$ maps the boundary $\partial D$ of $D$ into the boundary ...
1
vote
1answer
35 views

Identity involving the zeta function

This might be very trivial, but a proof I'm reading on the bounds of the zeta-function uses the following fact: If $s=\sigma+it$ is a complex number and if $\sigma\geq 2$, then $|\zeta(s)|\geq ...
0
votes
0answers
26 views

The amplitude for a photo from a short burst?

I'm imagining a monochromatic point source which is turned on for 1 second. If I take the Feynman propagator $G(x-y) = \frac{1}{(r^2-t^2+i \epsilon)}$ And I integrate it over time for one ...
1
vote
0answers
34 views

Automorphisms of the unit disk

I ran into a bit of a hurdle as I was reading a proof in Newman Bak's Complex Analysis regarding the form of automorphisms of the unit disk. The proof begins by showing that ...
1
vote
1answer
29 views

How can I visualize the interaction of the imaginary parts of the cosine/sine functions?

So I've been trying to get a good and intuitive feel for the extension of the sine and cosine functions into complex numbers (i.e. $\cos(z)$ where $z=a+bi$), and to do so I've naturally been looking ...
2
votes
2answers
40 views

domain of convergence of $\sum_{n=0}^{\infty}\left(\frac{1-e^z}{1+e^z}\right)^n$

Determine the domain of convergence of $$\sum_{n=0}^{\infty}\left(\frac{1-e^z}{1+e^z}\right)^n$$ Let , $$a_n=\left(\frac{1-e^z}{1+e^z}\right)^n$$ Then , $$\lim_n ...
1
vote
0answers
30 views

Complex integration over a simple pole

A paper I am reading addresses the following integral: $$\int^\infty_{-\infty}\frac{F'}{F}(1+it)h(t)dt$$ where $F$ is a function of $s\in\mathbb C$ with a simple pole at $s=1$, and $h$ is a smooth, ...
2
votes
2answers
29 views

Fixed points of $\frac{1\pm \sqrt{1-|a|^2}}{\bar a}.$

Prove that $\phi_a(z)=\frac{a-z}{1-\bar az}$ , $0<|a|<1$ has exactly two fixed points ; one inside the unit disc and the other outside the unit disc. Putting $\phi_a(z)=z$ I find that there ...
1
vote
1answer
34 views

Why are the exponents in the Schwarz-Christoffel mapping of the form (1- alpha/pi)?

Here is the Wikipedia article on it: https://en.wikipedia.org/wiki/Schwarz%E2%80%93Christoffel_mapping I feel it doesn't make sense. The integration produces real values (on the real line), so f(x) ...
2
votes
1answer
24 views

A question about complex integration

Let $z_0\in \Bbb C$ and $R>0$. Let $f:B(z_0,R)\to\Bbb C$ be a complex function such that $f=u+iv$ (where $B(z_0,R)$ is the open disc centered at $z_0$ with radius $R$). If $u$ and $v$ have ...
5
votes
1answer
57 views

Suppose $\sum_{k=-\infty}^{\infty}a_kz^k$ and $\sum_{-\infty}^{\infty}b_kz^k$ converge to $1/\sin(\pi z)$. Find $b_k-a_k$.

Suppose that the Laurent series $\sum_{k=-\infty}^{\infty}a_kz^k$ converges to $1/\sin(\pi z)$ when $0<|z|<1$, and suppose that the Laurent series $\sum_{k=-\infty}^{\infty}b_kz^k$ converges ...
5
votes
4answers
98 views

integration of $\int_0^{2\pi} cos^{2n}(t)dt$

Show that for any $n \in \mathbb{N}$, $$\frac{1}{2\pi}\int_0^{2\pi}\cos^{2n}(t)dt = \frac{1 \cdot 3 \cdot 5 \cdots(2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n}$$ To solve this problem, I was thinking that I ...
2
votes
2answers
67 views

Trig functions of complex numbers

I was studying complex numbers with the help of Boas textbook. I came about certain problems, which I solved only to find that the answers provided in the solution manual to be different. ...
3
votes
1answer
95 views

Computing residues of $\cot(\pi z)/z(z+1)$ with symmetries

I would like to know if there is a quick way of computing the residues of $$f(z) = \frac{\cot \pi z}{z(z+1)}$$at the points $z = 0$ and $z = -1$. They are double poles. Expanding this in Laurent ...
8
votes
0answers
164 views

Help with the integral $\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$

Referring to a previous question, i want help with the integral : $$\int_{0}^{\infty}\frac{\log(1\pm ix)^{2}}{\left(\frac{t}{2}\log(1 \pm ix) \right )^{2}-\pi ^{2}n^{2}}e^{-2\pi mx}dx$$ Where $n,m$ ...
3
votes
2answers
62 views

If $f$ has an essential singularity at $0$, there is a sequence $z_n \to 0$ such that $z_n^n f(z_n) \to \infty$

Here's a problem I was just working on: Let $f$ have an essential singularity at $0$. Show that there is a sequence of points $z_n \to 0$ such that $z_n^n f(z_n)$ tends to infinity. I know ...
3
votes
2answers
37 views

Help evaluating residue with simple poles

I am having a bit of trouble evaluating $$\sum_{k=1}^3{ \rm Res}\left(\frac{\log(z)}{z^3+8};z_k\right)$$ where $z_1=2e^{i\pi}$, $z_2=2e^{i\pi/3}$ and $z_3=2e^{i5\pi/3}$. I know that each $z_k$ is a ...
1
vote
0answers
24 views

Why are rational numbers required in cusps of congruence subgroups?

While we consider the action of congruence subgroups on $\mathbb{H}$ (the upper half plane), we compactify using an additional point at infinity, that is fine. But why do we add even all rational ...
1
vote
1answer
56 views

If $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Omega$ prove that $f$ and $g$ have the same number of zeros in $\Omega$

Let , $f$ and $g$ be analytic on $\Omega$ and continuous on $\Omega\cup \partial \Omega$. If $|f(z)+g(z)|<|f(z)|+|g(z)|$ on $\Omega$ prove that $f$ and $g$ have the same number of zeros in ...
-1
votes
0answers
33 views

Pointwise convergence to uniformly convergence on compact

Let $D \subset \mathbb{C}$ be a domain and $E$ be a subset of $D$ which has a limit point in $D$. And let $\{f_n\}$ be a sequence of holomorphic functions on D. Suppose that that $\{f_n\}$ converges ...
1
vote
0answers
18 views

A question about the differentiability conditions for a fractional power

Now consider the fractional power $$f(x)=z^{m/n}=|z|^{m/n}(\cos \frac{mArgz}{n}+i \sin \frac{mArgz}{n})$$ Since $f(x)$ satisfies the Cauchy-Riemann equations, and is therefore differentiable, we can ...
5
votes
1answer
137 views

Prove that $f$ is a polynomial

If $f(z)$ is an entire function and $|f(z)|\ge1$ for all $z$ with $|z|\ge \pi$ then show that $f$ is a polynomial. I tried to apply Lioville's theorem on $f$. For $|z|\le \pi$ , $|f(z)|\le k$ for ...
0
votes
1answer
33 views

$\frac{\partial u}{\partial x} \cos \theta + \frac{\partial u}{\partial y} \sin \theta = \frac{\partial u}{\partial r} (z_0 + r e^{i\theta})$?

Let $u$ be a function of two variable and all its partial derivative exists and fix $z_0 \in \mathbb C $ and $r>0.$ My vague question: How to show: $\frac{\partial u}{\partial x} \cos \theta + ...
3
votes
1answer
43 views

Showing pre-image under entire function is simply connected.

I am currently working on the following problem and have run into a bit of trouble: Consider an entire function $f$ s.t. $\overline{B_1(0)}\subset f(\mathbb{C}).$ Show that V, a component of ...
1
vote
0answers
26 views

Why only congruence subgroups for modular forms?

When we define modular forms, why do we restrict ourselves to congrumence subgroups? Why not any subgroup of finite index? Or, even more generally why not any subgroup? Is it just a matter of ...
1
vote
2answers
34 views

Schwarz reflection principle, understanding the conjugated function:

Given a symmetric region $\Omega$, say, symmetric w.r.t. the real line, and f(z) defined and analytic only on $\Omega^{+}$, we can analytically continue the function to $\Omega^{-}$ with the analytic ...
0
votes
1answer
27 views

Why does a conformal mapping create a full tiling of semi-infinite strips in the w-plane?

I know that, specifically for linear fractional transformations, symmetric points get mapped to symmetric points. So, if the real line gets mapped to a circle, then under a LFT, points symmetric ...
4
votes
0answers
35 views

Laplace transform of the logarithmic integral function

What is the Laplace transform of the logarithmic integral function $\text{li}(t)$. Meaning, how to compute the integral : $$\int_{0}^{\infty}\text{li}(t)e^{-st}dt$$
0
votes
0answers
23 views

One side half twist mobius band are one to one transformation on extended complex plane

I try to understand how one side half twist mobius are one to one transformation on extended complex plane $C^{*} = \mathbb{C} \cup \infty$ I don't ask for full proof, but I do want "something" ...
0
votes
0answers
30 views

Newman's proof of Prime Number Theorem - Zagier

The steps can be found here : http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf In his step two, it says that we need to show $\zeta(s) - \frac{1}{1-s}$ is holomorphic in ...
5
votes
0answers
40 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
2
votes
0answers
98 views

A difficult integral $\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$

Here is an integral that I want to see a different approach: $$\int_0^{\infty} \frac{\sin 2t}{1+t^3}\, {\rm d}t$$ Well, for someone who is deeply aware of the exponential integral function and the ...
3
votes
3answers
90 views

What is the value of $\sum e^{-n} \sin^2 n$?

Clearly the series $\sum_1^\infty e^{-n} \sin^2 n$ converges. If I put it into Maple, I get an exact value: $$ -\frac {{\rm e}^1 ( {\rm e}^1 (\cos(1))^2 + (\cos(1))^2 - {\rm e}^1 -1 ) }{-4{\rm e}^2 ( ...
1
vote
1answer
30 views

how to prove complex rational function has this property

Let's say we have a rational function $f$ (i.e polynom divided by polynom) , and assume that $f$ has no poles in the upper plane $\{z;Imz \geq 0\}$. we have to prove that: $$sup\{|f(z)|; Imz \geq ...
3
votes
3answers
55 views

Continuity of distance function

I wonder if this is obvious because it does not appear to me obvious at all: Reference: [Hormander: An introduction to Complex Analysis in Several Variables], page 37: Here is the quote Now, let ...
2
votes
1answer
32 views

Existence of an entire function with certain property

Let $\{a_n\}$ and $\{b_n\}$ be two sequence of complex numbers such that $|a_n|\to\infty$ as $n\to\infty$. Prove that there exists an entire function $f:\Bbb C\to\Bbb C$ (i.e. $f$ is complex ...
1
vote
2answers
96 views

map the UHP to an equilateral triangle

Explain how the upper half-plane can be mapped one-to-one and conformally onto an equilateral triangle. Thanks,
1
vote
2answers
44 views

Integration in a semicircle

Evaluate : $$\int_\Gamma(z^2+3\bar z)\,dz$$ where $\Gamma$ is the upper half of the unit circle from $-1$ to $+1$. First put , $z=e^{i\theta}, 0\le\theta\le\pi$ and take a negative sign as the ...
0
votes
1answer
38 views

Composition of analytic functions is analytic

I want to find a proof that shows the composition of two analytic functions is analytic. I know I should prove this using Cauchy-Riemann equations, but I wasn't able to use them in the proper way in ...
0
votes
1answer
18 views

Describe the Riemann surface for $w^2=z^2-1$.

Describe the Riemann surface for $w^2=z^2-1$. [Editorial note: I'm posting this to save an answer on another question, which actually answers this question instead.]
3
votes
2answers
460 views

Compute this integral, using a method other than the Residue Theorem,

$\int_0^\infty$ $\frac{1}{1+x}$$\frac{dx}{\sqrt{x}}$ Part (a) asks to compute the integral by means of the residue at x = -1. I have done this just now, and the answer is $\pi$. Part (b) asks, "can ...
1
vote
1answer
18 views

Bound on imaginary parts of family of analytic functions

Question: Let $ F$ be the set of holomorphic maps $f$ from the unit disc into the upper half plane, such that $f(0)=i$. Show that the supremum of the imaginary parts, $\sup_{f\in F}$ Im[$f(\frac i ...
2
votes
1answer
29 views

What does complexification mean in complex analysis, .e.g., in residue calculus,

I've learned complexification formally in a graduate linear algebra class. But what does the word mean in the setting of complex analysis? If I consider a real integral on the positive half line, ...