Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

21
votes
0answers
2k views

A difficult integral

For $\gamma>0,\delta>0$, trying to evaluate this integral: $$ I=\int_0^H\frac{e^{i t x} \log\left(\frac{H}{H-x}\right) ^{\frac{1}{\gamma }-1} \left(\left(\frac{k}{H \log \left(\frac{H}{H-x}\...
21
votes
0answers
863 views

A conformal mapping onto a region bounded by convex contours (Ahlfors)

I want to solve the following exercise (from Ahlfors' text, page 261) *3. Using Ex. 2, show that $p + q$ maps $\Omega$ in a one-to-one manner onto a region bounded by convex contours. Comments: (i)...
19
votes
0answers
329 views

Witt's proof of Gelfand-Mazur / Ostrowski's theorem

Now asked on MathOverflow. Background: It seems that, after his groundbreaking work on quadratic forms and inventing Witt vectors, Ernst Witt developed the hobby of giving extremely short proofs to ...
19
votes
0answers
496 views

What can be said about the level set of the real part of an analytic function?

I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which ...
17
votes
0answers
441 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 ...
13
votes
0answers
224 views

Is $\frac{1}{H(x) \pm i0}$ a distribution if $|\nabla H| \neq 0$ for $H(x)=0$?

I know that $\frac{1}{x \pm i0}$ is a tempered distribution in $\mathcal{S}'(\mathbb{R})$, see e.g. the Sokhotski–Plemelj theorem. In some lecture notes online I found the following statement (without ...
13
votes
0answers
445 views

Calculate using residues $\int_0^\infty\int_0^\infty{\cos\frac{\pi}2\Big(nx^2-\frac{y^2}n\Big)\cos\pi xy\over\cosh\pi x\cosh\pi y}dxdy,n\in\mathbb{N}$

Q: Is it possible to calculate the integral $$ \int\limits_0^\infty \int\limits_0^\infty\frac{\cos\frac{\pi}2 \left(nx^2-\frac{y^2}n\right)\cos \pi xy}{\cosh \pi x\cosh \pi y}dxdy,~n\in\mathbb{N}\...
12
votes
0answers
524 views

More elegant $\zeta(s)$ zeros counting function than $N(T)$

The explicit formula expresses the deep connection between the primes $p$ and the non-trivial zeros $\rho$ of $\zeta(s)$. The prime-counting function is given by the following formula giving primes in ...
11
votes
0answers
427 views

What differentiates algebraic geometry over $\mathbb{C}$ from complex analysis?

I have just begun learning algebraic geometry, and there are a lot more connections to complex analysis than I expected. For example: $\mathbb{CP}^1$ is the Riemann sphere Elliptic curves are ...
11
votes
0answers
695 views

Contour integral representation of Confluent Hypergeometric Function

My brain is spinning around in circles trying to reconcile three distinct contour-integral representation of the confluent hypergeometric function $_1F_1(a,b,z)$ for $b \in \mathbb{Z}_+$: From K.T....
10
votes
0answers
195 views

If a product of polynomials converges, does some product of their zeros also converge?

Suppose $\{p_k\}$ is a sequence of polynomials with $p_k(0)=1$. Let $a_1,a_2,\ldots$ be an enumeration of all of the zeros of the $p_k$. Suppose that $$\prod_{k=1}^\infty p_k(z)$$ converges ...
10
votes
0answers
300 views

How to generalize Reshetnikov's $\arg\,B\left(\frac{-5+8\,\sqrt{-11}}{27};\,\frac12,\frac13\right)=\frac\pi3$?

We have, $$\arg z_1 = \frac{k\,\pi}3, \quad z_1 = \left(\tfrac{1+\sqrt{-3}}{2}\right)^k\tag1$$ $$\arg z_2=\frac{k\,\pi}3, \quad z_2 = \left( B\Big(\color{blue}{\tfrac{-5+8\,\sqrt{-11}}{27}};\,\tfrac12,...
10
votes
0answers
214 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 ...
10
votes
0answers
823 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
10
votes
0answers
706 views

Contour integration with 2 branch points

I need to compute a quite complicated Fourier transform, but I'm having problems due to the facts that I have two branch points. The integral I need to solve is $$\int_\infty^{-\infty} \frac{dk}{\...
10
votes
0answers
746 views

Distribution of the sum of absolutes values of T-distributed random variables

Where X is a r.v. following a symmetric T distribution with 0 mean and tail parameter $\alpha$. I am looking for the distribution of the n-summed independent variables $ \sum_{1 \leq i \leq n}|x_i|$....
9
votes
0answers
269 views

A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
9
votes
0answers
236 views

Check whether a polynomial ideal is prime in the power series ring

I would like to know whether the ideal $I = \langle y^{2}(y^{2}-x^{2}) + w^{7}, y^{2}(y^{4}-x^{4}) + z^{7}\rangle$ is prime in $\mathbb{C}[[x,y,z,w]]$, the ring of formal power series in the ...
9
votes
0answers
628 views

Subadditivity for Analytic Capacity Disjoint Compacts separated by a Line

The following problem is asked in Greene and Krantz, Problem 9, page 382: Suppose that $C_1$ and $C_2$ are disjoint compact sets in $\mathbb{C}$ that can be separated by a line $l$ with $C_1 \cap l ...
9
votes
0answers
739 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] \frac{1}{...
9
votes
0answers
322 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 ...
8
votes
0answers
182 views

Using residue theorem to calculate following integral

I'm trying to evaluate the following integral using the residue theorem \begin{align}\label{eq:int_1} S(z) = \dfrac{1}{2\pi}\int_{0}^{2\pi} \dfrac{e^{i\phi}+z}{e^{i\phi}-z} e^{-\lambda\sin^{2}(\phi/2)...
8
votes
0answers
131 views

Is the normalized derivative of a holomorphic function Sobolev?

Let $B=\{z\in \mathbb C \,|\,|z|\le 1\}$ be the closed unit disk, and let $f:B \to \mathbb{C}$ be holomorphic. More precisely, I assume that $f$ is holomorphic on the interior $\text{int}(B)$, and ...
8
votes
0answers
149 views

Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually?

Is there any $a, b \in \mathbb R, b \ne 0$ such that $\Gamma(a+bi)$ can be evaluated manually? (Like $\Gamma(\frac 12)$) If there is/are, could you show me how to calculate it? I found that $\Gamma(...
8
votes
0answers
137 views

What is my method of rendering the Mandelbrot set called?

Recently I've dived into studying complex analysis on my own and it led me to find an interesting way of displaying the Mandelbrot set: Consider $$f(f(f(f(...x))))$$ Then, $$ \frac{d}{dx} (f(f(f(f(.....
8
votes
0answers
311 views

There exists polynomial $(P_{n})_{n\in\mathbb{N}}$ such that $P_n(0)=1$, and $\lim_{n\rightarrow \infty}P_{n}(z)=0$..

Show that there exists a sequence of polynomial $(P_{n})_{n\in\mathbb{N}}$ such that $P_n(0)=1$ for each $n$, and $\lim_{n\rightarrow \infty}P_{n}(z)=0$ for all $z\in \mathbb{C}\setminus \left\{0\...
8
votes
0answers
305 views

Complex Analysis with differential forms

I'm studying a little of Complex Anlysis and I have seen that I can thing the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n$. For example I know that Cauchy ...
8
votes
0answers
280 views

Fourier transform of the critical line of zeta?

Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line? I'd love to say that it's a weighted sum of delta distributions, ...
8
votes
0answers
322 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 $n,m$ ...
8
votes
0answers
134 views

How to recognize when a function is secretely holomorphic

Let $f : M \rightarrow N$ be a holomorphic map between complex manifolds (I'd be interested even in the case $M=N=\mathbb{C}$ which should not be much different). Now take $K$ a compact subset of $M$,...
8
votes
0answers
685 views

Interpretation of the Argument Principle

Recall that the argument principle states that given a meromorphic function $f$ and a compact region $K \subseteq \mathbb{C}$ whose boundary determines a simple contour and on which $f$ has no ...
8
votes
0answers
592 views

Separate incomplete elliptic integral into real and imaginary parts

I am working in a problem that involves Incomplete Elliptic Integrals of the First and Second kind of the form $F(\sin^{-1}x~|~m)$ and $E(\sin^{-1}x~|~m)$ where the parameters $m$, $x$ are real ...
8
votes
0answers
143 views

Weierstrass product expression for Klein's j-invariant

The first sentence of @ccorn's answer to a previous question of mine was: “Because of the modular symmetries of $j(\tau)$, the zeros of $j(\tau)$ are precisely the $\operatorname{SL}(2,\mathbb{...
8
votes
0answers
373 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
7
votes
0answers
79 views

When do polynomial equations come from complexification?

If $f(z) \in \mathbb{C}[z]$ is a polynomial of degree $d$, then it has $d$ complex zeros. Writing the complexification $$f(x+iy)=u(x,y)+iv(x,y)$$ we observe that the real polynomial system $u(x,y)=v(x,...
7
votes
0answers
62 views

Find the number of zeros of $f(z)=e^{z-1}-az$ inside unit disk, assuming $\mid a \mid >1$

This is an application of Rouche's theorem, I want to make sure I am doing it correctly: Let $f(z)=e^{z-1}-az$, where $\mid a \mid>1$ and $g(z)=-az$ Now, on the unit circle we have: $$\mid g(z) \...
7
votes
0answers
117 views

Solving a dual integral equation involving a zeroth-order Bessel function

Consider the following dual integral equations \begin{align} \int_0^\infty q^3 f_0(q) J_0 (qr) \, \mathrm{d} q &= g(r) \qquad\qquad\quad (0<r<1) , \\ \int_0^\infty f_0(q) J_0 (qr) \, \...
7
votes
0answers
107 views

Solution to Vector Lambert W function type Equation

I was wondering if anyone has any ideas for a closed-form solution to the equation $$Ax + \exp(x) +b =0$$ where $x,b \in \mathbb{R}^n$, $A$ is a symmetric positive definite matrix and $\exp$ denotes ...
7
votes
0answers
771 views

Good video lectures on complex analysis?

I would like to find a complete series of video lectures on complex analysis, preferably with the following conditions: The videos are in English and clearly recorded. (Using English as a second ...
7
votes
0answers
85 views

Are Differential Structure of a Complex Number and its Field Structure Independent?

It is well known that $\mathbb{R}^2$ has a very famous field structure defiend by $(a,b)(c,d)=(ac-bd,ad+bc)$. And it also has a holomorphic structure, which makes $z\mapsto z$ differentiable but not $...
7
votes
0answers
335 views

Closed form of $\sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!}$

Is it possible to calculate the sum $$ \sum _{n=0}^{\infty} \frac{\left(-\pi ^2\right)^n \cos \left(2^nb\right)}{(2 n)!} $$ in closed form? Formal naive argument gives $$ \sum _{n=0}^{\infty} \...
7
votes
0answers
218 views

Prove this fact about holomorphic functions.

Suppose $f \colon \mathbb{D} → \mathbb{C}$ is holomorphic. Then I want to show that the diameter $$d=\sup _{z, w∈\mathbb{D}} |f (z) − f (w)|$$ of the image of $f$ satisfies $2|f′(0)| ≤ d$ and that ...
7
votes
0answers
194 views

Complex 'mean-value-theorem'-like property implies quadratic

One of my friend asked me the following problem: Problem. Suppose that $f$ is a holomorphic function on a convex open set $U$ which satisfies the following property: For all distinct $z, w \in U$, ...
7
votes
0answers
235 views

Holomorphic extensions of characteristic functions

Let $\chi(\xi) := \mathbb{E}e^{i \xi X}$, $\xi \in \mathbb{R}^d$, be the characteristic function of a random variable $X$. It is widely known that $\chi$ admits a holomorphic extension to the strip $\{...
7
votes
0answers
399 views

Analytic function defined in the punctured unit disk real in the unit circle then $f(z)=\overline{f(1/\overline{z})}$

This is a follow up to a question I asked a few days ago. I initially thought I understood the solution to the problem, but there's something I can't quite grasp: Let $f$ be analytic in the set $\{ ...
7
votes
0answers
163 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
7
votes
0answers
272 views

Compactly supported Dolbeault Cohomology: is this True?

nLab states that for $D$ the unit disk in $\mathbb C$, the cohomology of the complex $$ (\Omega_c^{1,\ast}(D),\overline{\partial})$$ is the continuous dual of the space of holomorphic functions $\...
7
votes
0answers
127 views

Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
7
votes
0answers
689 views

Fejer-Riesz Lemma

I'm trying to apply the Fejer-Riesz Lemma constructively. The lemma says that for a Laurent polynomial $a(z) = \sum_{-n}^na_jz^j$ with $a_j = \bar a_{-j}$ and $a(e^{i\theta})\geq0$ on the complex unit ...
7
votes
0answers
361 views

One-dimensional projective group in linear transformation

A convenient way to express a linear transformation is by use of homogeneous coordinates. If we write $z=z_1/z_2$ and $w=w_1/w_2$ we find that $w=Sz$ if $$w_1=az_1+bz_2\text{ and } w_2=cz_1+dz_2$$ ...