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, ...

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15
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582 views

About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$

It is an interesting exercise to show that the function $f(x)=\sin(x\log x)$ is Riemann-integrable over $\mathbb{R}^+$ (as shown by robjohn in this related question, for instance). Even more ...
11
votes
0answers
432 views

Integral $\int_0^\infty \frac{\log^2 x \cos ax}{x^n-1}dx$

Hi I am trying to calculate $$ I:=\int\limits_0^\infty \frac{\log^2 x \cos (ax)}{x^n-1}\mathrm dx,\quad \Re(n)>1, \, a\in \mathbb{R}. $$ Note if we set $a=0$ we get a similar integral given by $$ ...
10
votes
0answers
264 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\bigl(\frac{1}{e^n}\bigr)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
9
votes
0answers
161 views
+50

Subring of $\mathcal O(\mathbb C)$

Let $\mathfrak A \subset \mathcal O(\mathbb C)$ be the subring generated by the nowhere zero analytic functions $f: \mathbb C \to \mathbb C$. Does we have a precise description of $\mathfrak A$ ? Is ...
9
votes
0answers
153 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and ...
9
votes
0answers
244 views

Conjecture: Tract version of Gauss--Lucas Theorem for higher derivatives.

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
8
votes
0answers
121 views

Integral Asymptotics for inhomogenous phase

I'm looking for asymptotics for an integral of the form: $$F(n):=\int_{1/2-i\infty}^{1/2+i\infty} e^{\phi(n,z)}dz$$ where $\phi(n,z)=(n-n^3)\log(1-z)+n^2\log(1+z)-n\log(z)$. One can solve for the ...
8
votes
0answers
334 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 ...
8
votes
0answers
235 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 ...
8
votes
0answers
412 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 would ...
7
votes
0answers
146 views

Work of Ted Kaczynski

I hope this question is not too crazy sounding, but I was wondering if anyone is familiar with the work of Ted Kaczynski (or even has cited/used it before). After reading in Lars Ahlfors' Complex ...
7
votes
0answers
99 views

Multivalued Functions for Dummies

I have been studying complex analysis for a while, but I still cannot "get" how multivalued functions work. Despite having it explained to me many times, my brain cannot process it. For example, I ...
7
votes
0answers
49 views

Decay of amplitude integral

Consider the function $$ f(\vec{x}) = \int_{\Bbb R^3} {\frac{ e^{-i\,\vec{x}\cdot\vec{k}}}{\sqrt{\vec{k}^2 + m^2}}} d^3 k $$ from Zee's Quantum Field Theory in a Nutshell. He argues like this: ...
7
votes
0answers
141 views

A problem about $e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$

Let $\alpha_i\in [0,1),\; i\in \{1,\cdots,N\}$ for some positive integer $N$, such that $$e^{2\pi i \alpha_1}+e^{2\pi i \alpha_2}+\cdots+e^{2\pi i \alpha_N}=0$$ and if for any non-empty proper subset ...
7
votes
0answers
118 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 ...
7
votes
0answers
122 views

Can a Power Series tell when to stop?

The naive description of the radius of convergence of a complex power series is as the largest radius so that the ball avoids poles and branch cuts. This makes sense in a world where analytic ...
7
votes
0answers
133 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
312 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
7
votes
0answers
235 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 $$w_1=az_1+bz_2\text{ and } w_2=cz_1+dz_2$$ ...
7
votes
0answers
153 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] ...
7
votes
0answers
262 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
64 views

Mittag-Leffler Expansion

I am attempting to perform what is described in my notes as a "Mittag-Leffler Expansion", but first must prove that this expansion is valid. Given that $$ f(z) = \frac{1}{\sin{z}} - \frac{1}{z}$$ ...
6
votes
0answers
151 views

Complex contour integral: How does the stationary point method used in this case?

I was reading a paper which has the following integral in order to do the inverse Laplace transformation: $$ I=\frac{1}{2\pi i}\int_{-i\infty+\gamma}^{i\infty+\gamma} ...
6
votes
0answers
143 views

estimating a particular analytic function on a bounded sector.

Let $f(z)$ be an analytic function on $C^+=\{\Re z>0\}$, and we have the following (weaker) estimates $$ |f(re^{i\theta},a)|\leq C (r\cos\theta)^{-n}, ...
6
votes
0answers
55 views

Methods for “recognizing” a polynomial of several variables.

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial. 1) Show that $f^{(n)}\equiv0$ for some $n\geq0$. 2) ...
6
votes
0answers
103 views

Proving that two functions involving integrals with Legendre polynomials are equal

I have two functions that I expect to be equal (where $P_{2l}$ are the even Legendre Polynomials): $$F_{2l}(x)=x\, \tanh(\pi x/2)\left|\int_0^1 u^{i x-1}P_{2l}(u)\,du\right|^2$$ ...
6
votes
0answers
87 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 ...
6
votes
0answers
163 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 ...
6
votes
0answers
386 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 ...
6
votes
0answers
233 views

If $\Re(f)$ is bounded then f is constant.

I have to solve following problem If $\Re (f)$ is bounded above or below for a function $f$ holomorphic on $\mathbb{C}$ then $f$ is constant. My attempt: If there is $M$ such that $\Re(f) \le ...
6
votes
0answers
241 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 ...
6
votes
0answers
1k views

Inverse Laplace Transform as Bromwich Integral

I am seeking a references that provide a rigorous treatment of the inverse Laplace transform (Bromwich integrals), and how to compute them (beyond using tabled solutions - they don't cover my needs, ...
6
votes
0answers
77 views

“Natural” interpolation between partial sums of a power series

Suppose $f(z)=\sum_{n=0}^\infty a_n z^n$ has a radius of convergence of $R$. Let the $N$-th partial sum be $f_N (z)=\sum_{n=0}^N a_n z^n$. What smooth (analytic) function interpolates between ...
6
votes
0answers
241 views

Identity involving $\zeta(3)$

This is related to this question and my partial answer to it. I've found that the proof of the 2nd identity reduces to showing that ...
6
votes
0answers
140 views

What is the complex *algebraic* moduli of elliptic curves?

It's well-known that $SL_2(\mathbb{Z}) \backslash \mathfrak{h}$ is a coarse moduli space for complex elliptic curves. Thus, I would expect this to be related to the pullback of $\mathcal{M}_{ell} ...
6
votes
0answers
378 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
375 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 ...
6
votes
0answers
2k 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 ...
5
votes
0answers
64 views

Evaluate $\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$ using residue calculus

I'm asked to evaluate $$\int_{-\infty}^{\infty}\frac{\sqrt{a+ix}}{a^2+x^2}\,dx$$ $\mathbb{R}\ni a>0$, using residue calculus (where $\sqrt{\cdot}$ is the PV $\sqrt{}$). My approach is as follows: ...
5
votes
0answers
106 views

What is reflection across parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
5
votes
0answers
54 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
5
votes
0answers
60 views

Show that the integral of Riemann function is analytic

I'm trying to resolve this problem. Let $\Omega$ be an open set no empty of $\mathbb C$, $[a,b]$ a compact interval of $\mathbb{R}$, further $f,\ g\colon[a,b] \to \mathbb C$ two integrable Riemann ...
5
votes
0answers
96 views

Integrating $\int_{-\infty}^\infty \frac{1}{1 + x^4}dx$ with the residue theorem

Calculate integral $$\int\limits_{-\infty}^{\infty}\frac{1}{x^4+1} dx$$ with residue theorem. Can I evaluate $\frac 12\int_C \dfrac{1}{z^4+1} dz$ where $C$ is simple closed contour of the upper ...
5
votes
0answers
56 views

Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
5
votes
0answers
97 views

Generating function of the squared Riemann zeta function

It's a well known fact that $$\sum_{k=2}^{\infty} \zeta(k) x^k=-x \psi(1-x)-x\gamma \space (|x|<1) $$ but I didn't meet yet a version for squared Riemann zeta function $$\sum_{k=2}^{\infty} ...
5
votes
0answers
153 views

A Curious Identity

I met the following equations when I was trying to solve a complex line integral (W.Rudin, RCA, p.228 ex.13). My question is how to prove them: We have to show that for $n>2$ even $$ ...
5
votes
0answers
719 views

contour integration of a function with two branch points .

Many of us have seen the evaluation of the integral $$\int^{\infty}_0 \frac{dx}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$$ It can be solved using contour integration or beta function . I thought of ...
5
votes
0answers
175 views

Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
5
votes
0answers
71 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
5
votes
0answers
191 views

finding Laurent expansion of a periodic function

How are Laurent series and Fourier series related to each other? There is a problem that states that for a periodic function $F(z + 2 \pi ) = F(z)$ that is analytic in finite plane. $$F(z) = ...