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.

10,668 questions with no upvoted or accepted answers
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33
votes
0answers
497 views

Differential "Freshman's dream" for Laplacian operator.

Today I encountered quite an interesting phenomenon. There is an exercise in multivariable calculus that asks students to prove the identity $$ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\...
32
votes
0answers
1k views

Evaluate $\int_{-\infty}^\infty\frac{x}{\sin^2(\sqrt{x})\sinh^2\left(2\sqrt{2x}\right)+\pi^2\cos^2(\sqrt{x})\cosh^2\left(2\sqrt{2x}\right)}\mathrm dx$

I encountered an astonishing integral (numerically verified): $$\int_{-\infty}^\infty \frac{x\ \mathrm dx}{\sin ^2(\sqrt{x}) \sinh ^2(2 \sqrt{2 x})+\pi ^2 \cos ^2(\sqrt{x}) \cosh ^2(2 \sqrt{2 x})}\\ \...
27
votes
0answers
621 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 ...
26
votes
0answers
980 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)...
23
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0answers
620 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 ...
22
votes
2answers
1k views

How to find area of a polygon built on the roots of a given polynomial?

How to find the area of a (maximum area convex) polygon, built on the roots of a given polynomial in the complex plane? For example, consider the equation: $$2x^5+3x^3-x+1=0$$ It has one real and four ...
20
votes
0answers
507 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 ...
16
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0answers
417 views

How large must $f(f(z))-z^2$ be in the unit disk?

This is a follow-up question to $|f(f(z))-z^2|$ must be large somewhere in the disc $\mathbb{D}$?, where the following was proven: Let $f$ be holomorphic in the unit disk $\Bbb D$ with $f(0) = 0$ and ...
15
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0answers
2k 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}\...
14
votes
0answers
408 views

Jordan Curve Theorem, Professor Tao's proof

Here is Professor Terry Tao's proof of the Jordan curve theorem using complex analysis, I more or less followed the proof until the following paragraph (see section 4). (Actually there is no need to ...
14
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0answers
228 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(...
14
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0answers
261 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 ...
14
votes
1answer
249 views

Eigenvalue problem for $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue $E$ in the complex plane of $x$ for one dimensional Schrodinger equation $$ −\psi''(x) − (ix)^ N \psi(x) = E\psi(x). $$ where $N$ can be any real number, the boundary condition ...
14
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0answers
1k 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.
13
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0answers
738 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 ...
13
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0answers
495 views

Complex Analysis with differential forms

I'm studying a little of Complex Analysis and I have seen that I can use the integrals of complex functions as integrals of differential forms in $\mathbb{R}^n.$ For example: Cauchy Theorem for ...
11
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0answers
678 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
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0answers
391 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
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0answers
150 views

Roots of partial sum of power series

Consider the power series $$ \sqrt{1+z} = \sum_{k=0}^\infty \left( \begin{array}{c} \frac{1}{2} \\ k \end{array} \right) z^k. $$ I am interested in characterizing the roots of the partial sum $$ s_n(z)...
10
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0answers
446 views

For $n\in\mathbb{N}^+$, $\Re(s)>0$, evaluate $\int_{0}^{\infty}\sin\left(2\pi ne^{x}\right)\left[\frac{s}{e^{sx}-1}-\frac{1}{x}\right]dx$

Let $n$ be a positive integer, and $s\in \mathbb{C}\;,\Re(s)>0$. I want to compute the integral : $$\int_{0}^{\infty}\sin\left(2\pi ne^{x}\right)\left[\frac{s}{e^{sx}-1}-\frac{1}{x}\right]dx$$ I ...
10
votes
0answers
199 views

Find radius of convergence for a complicated series for $f'(f(x)) = f(f'(x))$

Several months ago, I answered this question asking for solutions to the functional equation $f'(f(x)) = f(f'(x))$ by expanding as a formal Taylor series around some arbitrary fixed point of $f$. This ...
10
votes
0answers
216 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
338 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, ...
10
votes
0answers
253 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
273 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
10
votes
0answers
674 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 ...
10
votes
0answers
760 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
1answer
967 views

How to choose contour in $\mathbb{C}$ to do Residue Integration.

I'm almost sure that there's not any simple way to answer this question, but I'll try. I'm studying complex variables and the method of calculating improper integrals with residues but I'm struggling ...
9
votes
0answers
181 views

When exactly is the principal AGM equal to the optimal AGM?

Definitions Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
9
votes
0answers
139 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) \...
9
votes
0answers
252 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(.....
9
votes
1answer
1k views

Uniform limit of one-to-one analytic functions is either constant or one-to-one

Let $U$ be a complex domain, and $(f_n)_{n\in \mathbb{N}}$ be a sequence on one-to-one analytic functions defined on $U$. Suppose that $f_n$ converges to $f$ uniformly on every compact subset of $U$. ...
9
votes
0answers
290 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
537 views

Computing the Fourier transform of the distribution $\|x\|^{-\alpha}$.

Question: Suppose we are given the tempered distribution $\|x\|^{-\alpha}$. We want to compute the Fourier transform $\mathcal{F}[\|x\|^{-\alpha}](\xi)$. What techniques are available for ...
9
votes
0answers
282 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
1k views

"Angle-preserving" equivalent to conformal?

I'd like to investigate the common turn of phrase that conflates "angle-preserving map" with "conformal map". Let $f:\Bbb R^2\to\Bbb R^2$ be a continuous function. I'll define $f$ to be angle-...
9
votes
0answers
179 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{...
9
votes
0answers
869 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
337 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
104 views

Regarding definition of Linear Convexity in $\mathbb{C}^n$ and reference request

In the book Notions of Convexity by Lars Hörmander page 290, section 4.6 Linear convexity is defined as follows. An open set $X\in \mathbb{C}^n$ is called linearly convex if for every $z\in \mathbb{C}^...
8
votes
0answers
114 views

Is an entire function "determined by" its maximum modulus on each circle centered at the origin?

Let $f$ be an entire function, and $$M_f(r)=\max_{|z|\leq r}|f(z)|$$ denotes its maximum modulus on the circle centered at the origin with radius $r>0$. It's clear that for any entire functions $f(...
8
votes
0answers
101 views

When do differential equations induce maps between algebraic varieties and how to find these varieties

The intuition: Consider a single variable polynomial differential equation with integer-polynomial coefficients, for example $$ y '' = -y $$ Then consider a pair of algebraic varieties (that is ...
8
votes
0answers
150 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
108 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,...
8
votes
0answers
383 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
395 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 $\{...
8
votes
0answers
375 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
142 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
839 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
420 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 ...

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