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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288 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 ...
10
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233 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 ...
10
votes
0answers
387 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 $$ ...
9
votes
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144 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
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232 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
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115 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
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219 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
399 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
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110 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
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101 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
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129 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
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243 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
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224 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
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307 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 ...
7
votes
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151 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
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259 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
51 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
123 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
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138 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
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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
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101 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
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82 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
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325 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
209 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
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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
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231 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
351 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
346 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
72 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 ...
5
votes
0answers
73 views

the real part of a holomorphic function on C \ {0, 1}

Let $h$ be a real valued harmonic function on the twice punctured plane $Ω = \text{C \ {0, 1}}$. Show that there exist unique real numbers $a_0$, $a_1$ such that $u(z) = h(z) − a_0 \log |z| − a_1 \log ...
5
votes
0answers
69 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
5
votes
0answers
67 views

Solve complex equation with exponential

I have to solve: $$e^{3z}+3ie^{2z}-ie^z+3=0$$ My attempt: Let $0\ne x:=e^z$. Then we can rewrite our equation as: $$x^3+3ix^2-ix+3=0$$ $$ix^2(-ix+3)+(-ix+3)=0$$ $$(-ix+3)(ix^2+1)=0$$ So $x\in ...
5
votes
0answers
57 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
55 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
94 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
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0answers
158 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 ...
5
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149 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
686 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
70 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
134 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} ...
5
votes
0answers
78 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
165 views

Is there a closed form expression for this integral?

I've been trying to find a closed form expression/series expansion for the following integral without success: $$F(a,b)=\int_{\epsilon-i\infty}^{\epsilon+i\infty} ...
5
votes
0answers
150 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
183 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
149 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 ...
5
votes
0answers
98 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
399 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
163 views

History Question - Branch Cut

My professor began discussing branch cuts in class today and mentioned that he did not know the origin of the term. Does anyone know the origin of the term and perhaps a source that talks about it?