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, ...
28
votes
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568 views
Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?
As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in ...
9
votes
0answers
86 views
Is there a classification of isolated essential singularities?
In the thread Why do we categorize all other (iso.) singularities as "essential"?, here is one of the questions that was asked:
Do we not care about essential singularities to classify ...
9
votes
0answers
160 views
Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
7
votes
0answers
194 views
On the continuation of a polynomial
This exrcise is from the first section of Marden:
Exercise 12. Let the interior of a piecewise regular curve $C$ contain the origin $\cal O$ and be star-shaped with respect to $\cal O$. If the ...
6
votes
0answers
148 views
Hilbert transform and Hilbert matrix
The Hilbert matrix is
\begin{bmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt]
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt]
...
6
votes
0answers
226 views
an infinite series expansion in terms of the polylogarithm function
we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though ...
6
votes
0answers
109 views
$L^{2}(\mathbb R)$- norm of entire function
Let $f(z)$ be an entire function defined by $$f(z)=\prod_{n=1}^{\infty}\bigg(1-\frac{z^{2}}{a_{n}^{2}}\bigg),\qquad z\in \mathbb C$$
where $\{a_{n}\}_{n=1}^{\infty}$ is a sequence of positive real ...
6
votes
0answers
155 views
Dolbeault cohomology of the complex projective space.
Let $X=\mathbb{CP}^n$. We proved using the hodge decomposition that $H^0(X,\Omega^p)=0$ if $p\neq 0$. But I do not understand why I cannot have global holomorphic differential p-forms not even ...
5
votes
0answers
76 views
counterexample to Abel's theorem
$D=\{z\in\Bbb C:|z|<1\}$. Let $f(z)=\sum\limits_{n=0}^\infty a_n z^n(a_n,z\in\Bbb C)$ be a power series, the radius of convergence of $f$ is $1$, $\sum\limits_{n=0}^\infty a_n =s$.
Give an ...
5
votes
0answers
62 views
$\int_{-\infty}^{+\infty}\left|f(g(t))-f(h(g(t)))\right| \,dt=0$ How to find $f$?
Let $h,g$ be given entire functions.
Consider
$$\int_{-\infty}^{+\infty}|f(g(t))-f(h(g(t)))|\, dt=0,$$
where $|\cdot|$ means modulus.
How do I find non-polynomial analytic solutions for $f\,$?
...
5
votes
0answers
92 views
Linear algebra estimates
Here is something that has been troubleing me lately. I don't know if it is true of not. I suspect it is.
Suppose that $A,B$ are two $n \times n$ matrices with complex entries. $A^t = A$, $\bar B^t = ...
5
votes
0answers
149 views
Primality using $\Gamma(x)$
Wilson's theorem states $n \in \mathbb N$ is prime iff$(n-1)! \equiv -1\pmod n$.
$\Gamma$-function extends the usual factorial to complex numbers.
What are the complex numbers such that $\Gamma(z)+1 ...
5
votes
0answers
165 views
Exponential type functions
An entire function $f(z)$ is of exponential type $\alpha$ if there exists $A$ such that $|f(z)|\leq Ae^{\alpha|z|}$ for all $z\in \mathbb C$. Given that $A=1$: how to prove that ...
5
votes
0answers
83 views
Showing that $[\mathbb C(z):\mathbb C(f_0)]=$topological degree of $f_0\in\mathbb C(z)$ as a function from the Riemann sphere to itself
McKean & Moll offer the following sketch of a proof.
Let $P(x)=a_0(x)-f_0b_0(x)$ in $\mathbb C(f_0)[x]$ where $f_0(z)=\dfrac{a_0(z)}{b_0(z)}$ is in $\mathbb C(z)$. Then evidently, $\deg P=\#$ ...
5
votes
0answers
277 views
An argument on page 62 of Griffith's book, “Introduction to Algebraic Curves”
I am a bit confused about some of the things that Griffiths says on page 62 of his book, Introduction to Algebraic Curves. I am not sure how I can reproduce the text here. I can see that GoogleBooks ...
5
votes
0answers
579 views
Choosing the branch of a logarithm
The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
4
votes
0answers
40 views
Complex differentiable implies analytic
The only proof of this that I am aware of, uses complex line integration. This tool allows for elegant proof, but has deprived me of any intuitive or geometric understanding of why this is the case. ...
4
votes
0answers
47 views
When are two commuting linear operators functions of each other
I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up.
If we formally consider the integral operator $E ...
4
votes
0answers
30 views
Conformal sets in $\Bbb C^n$
I'm trying to show whether these two sets are conformally equivalent.
$\Delta_n=\{(z_1,\dots,z_n):|z_i|<1,1\le i\le n|\}$ and $\Omega=\{(z_1,\dots,z_n):\text{Im}(z_1)>0\}$ where $n\ge 2$Thank ...
4
votes
0answers
106 views
extension of Cauchy's Integral formula
This question is from Brown and Churchill's Complex Variables and Applications, 8ed., Section 52, Question 6.
Let $f(s)$ denote a continuous function taken along a simple contour, $C$ enclosing a ...
4
votes
0answers
39 views
number of zeros of complex waves
Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
4
votes
0answers
75 views
When does a modular form satisfy a differential equation with rational coefficients?
Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ for $\Gamma$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least ...
4
votes
0answers
97 views
Möbius transformations form a simple group
How to show the group $M$ of Möbius transformations is a simple group?
I know: $SL_2(\mathbb C)/\{+I,-I\}\cong M$ then if $A \lhd M \implies \phi^{-1}(A) \lhd SL_2(\mathbb C)/\{+I,-I\}$.
So if ...
4
votes
0answers
55 views
adding a second parameter to an integral
I came across an evaluation of $\displaystyle \int_{-\infty}^{\infty} \frac{\cos (ax^{2}) \cosh (ax)}{\cosh (\pi x)} \ \mathrm{d}x = \cos \left(\frac{a}{4} \right) \ \ (|a| \le \pi$) in a textbook.
...
4
votes
0answers
72 views
An entire function with finite covering group is a polynomial.
Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
4
votes
0answers
56 views
Invertibility of a Toeplitz operator
Let $\phi$ be a real-valued function. I am trying to show that the Toeplitz operator $T_\phi$ is invertible if the function 1 is in the range of $T_\phi$. Here is what I got so far:
There exists a ...
4
votes
0answers
84 views
invariant subspace of a Hardy space
Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
4
votes
0answers
183 views
contour integral with singularity on the contour
I want to compute the following integral
$$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$
The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
4
votes
0answers
39 views
Is the inverse of any elementary function asymptotic to some elementary function?
Is the functional inverse of any elementary function asymptotic to some elementary function ?
For instance Lambert's $W(z)$ is asymptotic to $ln(z)$. See ...
4
votes
0answers
38 views
A question on Analytic pertubation Theory
I am currently trying to understand "Analytic Pertubation Theory for Linear Operators" by T. Kato. In Chapter VII.§2.1 (p.375f) the following situation is considered:
Setup: Let $D \subset ...
4
votes
0answers
77 views
Constructing Riemann surfaces using the covering spaces
In the paper "On the dynamics of polynomial-like mappings" of Adrien Douady and John Hamal Hubbard, there is a way of constructing Riemann surfaces. I recite it as follow:
A polynomail-like map ...
4
votes
0answers
215 views
Confused by a proof in Rudin *Functional Analysis*
I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial.
...
4
votes
0answers
309 views
Complex Analysis - Contour Integral around $\frac{1}{\sin(z)}$
A function is given:
$$f(z)=\frac{1}{\sin(z)}$$
which has singular points along the real axis at $z=\pi n$ with integer $n$.
The residue at $z=\pi n$ is equal to $(-1)^{n}$ as can be computed using ...
4
votes
0answers
243 views
What are applications of Lagrange's identity?
I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then
$$
\left|\sum_{i=1}^na_ib_i\right|^2 = ...
4
votes
0answers
58 views
What is the term used for space of analytic functions?
I deal with analytic functions in the unit disc represented as the series
$\sum_{n=0}^\infty u_n z^n$, where the coefficients $u_n$ satisfy the condition
$\sum_{n=0}^\infty n^\alpha|u_n| < \infty$ ...
4
votes
0answers
63 views
Convex pentagons are similar if conformally equivalent.
The problem:
Suppose two convex pentagons $A$ and $B$ have equal interior angles (that is, $A=A_1A_2A_3A_4A_5$ and $B=B_1B_2B_3B_4B_5$) with $\angle A_j =\angle B_j$ for each $j\in\{1,\ldots,5\}$).
...
4
votes
0answers
240 views
Why is every conformal bijection between disks a linear fractional transformation?
Why is every conformal bijection between disks actually a linear fractional transformation?
I thought I could justify this claim with the following idea.
Suppose $f$ is a conformal bijection from a ...
4
votes
0answers
508 views
Finding general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$.
I'm trying to find the most general harmonic polynomial of form $ax^3+bx^2y+cxy^2+dy^3$. I write this polynomial as $u(x,y)$.
I calculate
$$
\frac{\partial^2 u}{\partial x^2}=6ax+2by,\qquad ...
4
votes
0answers
108 views
Local uniformity implies Riemann integrability
Theorem.
Let $f_{n}:G\to \mathbf{C}, n\in \mathbf{N}$, continuous and integrable and let $f=\lim_{n\to \infty}f_{n}$ be locally uniform in $G$. Then $f$ is continuous and integrable in $G$.
...
4
votes
0answers
84 views
Converse to the Koebe Distortion?
Given a univalent function on the disk satisfying $f(0)=0$ and $f'(0)=1$, Koebe Distortion theorem says that
\begin{equation}
\frac{1-|z|}{1+|z|}\le \left|z\frac{f'(z)}{f(z)}\right|\le ...
4
votes
0answers
198 views
please help with the a gamma function since i don't even have the idea?
How to prove:
$$\frac{1}{2\pi i}\int_{-i\infty}^{i\infty} \frac{\Gamma(\alpha_1+x)}{\beta_1^{\alpha_1+x}}\, \frac{\Gamma(\alpha_2-x)}{\beta_2^{\alpha_2-x}}\, ...
4
votes
0answers
203 views
Visualizing the domain of the square root
I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this?
I would like some sort of ...
4
votes
0answers
192 views
Computing complex principal value integral - sgn-function?
I currently face a less appealing integral which emerged computing the expectation of some random variable. It reads as (omitting all unnecessary constants except $\alpha\in(0,1)$)
$$ PV ...
4
votes
0answers
234 views
Organizing types of functions by their calculus-related properties, in diagram form?
Does anyone know of a diagram that displays and organizes categories of functions according to their calculus-related properties (e.g. continuous, $C^\infty$, degrees of differentiability and ...
4
votes
0answers
133 views
When functions, analytically continued, carry over certain properties
Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
4
votes
0answers
134 views
Conformal structure of regions of the complex plane and the ring of holomorphic functions
How is the conformal structure of regions of the complex plane determined by the integral domain of holomorphic functions defined on those regions?
Thanks
4
votes
0answers
149 views
What are the properties of $\Re f(x)$ for an analytical function $f$ if $\Im f(x)\ge 0 \forall x\in\mathbb R_0^+$?
Background: In Electrodynamics, the scalar permittivity $\epsilon(\omega)$ relates the Electric displacement field $\vec D$ to the electric field $\vec E$ as $\vec D=\epsilon\vec E$ when assuming a ...
4
votes
0answers
348 views
Logarithm of a complex number as intersections of two logarithmic spirals
In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these ...
4
votes
0answers
257 views
The Milnor Conjecture on the Unknotting Number of a Torus Knot
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = ...
3
votes
0answers
59 views
How do zeros on the complex plane affect the real number line?
Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
