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, ...

learn more… | top users | synonyms (2)

16
votes
0answers
274 views

Is there an exposition of complex analysis firmly separating the algebra, analysis, and topology?

Complex analysis seems to work because of the interplay between algebraic geometry over $\mathbb{C}$, and analysis and topology exploiting the fact that $\mathbb{C}/\mathbb{R}$ happens to be a ...
11
votes
0answers
313 views

Proof of Cauchy's Beta Integral $\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}$

The Cauchy's Beta Integral is given by $$\int_{-\infty}^\infty \frac{dt}{(1+it)^x(1-it)^y}=\frac{\pi 2^{2-x-y}\Gamma(x+y-1)}{\Gamma(x)\Gamma(y)}$$ I would like to know how it is proved.
10
votes
0answers
227 views

Integrating $\int_0^\infty\frac{\log (1+z^2)}{e^z-1}dz$ using residue calculus.

I've been looking at how to integrate the following definite integral using the residue calculus, but can't seem to get my thoughts together. I know the $\log$ term is a multivalued function and the ...
10
votes
0answers
216 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
8
votes
0answers
41 views

Weierstrass's elliptic $\wp$-function

For $\omega_1,\omega_2 \in \mathbb{C}$ we define $\Gamma:=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2$ and $\Gamma':=\Gamma\setminus\{0\}$. $\wp$ is weierstrass's elliptic $\wp$-function ...
8
votes
0answers
85 views

Conceptual proof relating linear fractional transformations to matrices

Define a map from $2 \times 2$ invertible matrices to linear fractional transformations $$ f:\left( \begin{array}{ccc} a & b \\ c & d \\\end{array} \right) \mapsto \frac{az + b}{cz + d}.$$ ...
8
votes
0answers
162 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 $$w_1=az_1+bz_2\text{ and } w_2=cz_1+dz_2$$ ...
8
votes
0answers
223 views

Help in calculating the following integral $\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $

I was asked to calculate this: $$\int_0^{2\pi}\! \frac{(1+2\cos x)^n \cos (nx)}{3+2\cos x} \, \mathrm{d}x. $$ My idea was to change the integration limits to $|z|=1$ in the complex plane and to ...
8
votes
0answers
128 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. ...
7
votes
0answers
149 views

Simpler way to evaluate the Fourier transform of $\exp\left(i e^x\right)$?

I have the task to evaluate $|a(k)|^2$ with $$ a(k) = \int_{-\infty}^\infty \!dx\,\exp\left(i k x + i e^{x}\right).\tag{1}$$ The integral in (1) can be evaluated explicitly via the substitution ...
7
votes
0answers
189 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
0answers
316 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 ...
7
votes
0answers
233 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
70 views
+250

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
6
votes
0answers
73 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
234 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] ...
5
votes
0answers
66 views
+100

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 ...
5
votes
0answers
111 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
81 views

Evaluating $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k \sinh (k \pi)}$

You can evaluate $\displaystyle \sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{4n-1} \sinh (k \pi)}$ ($n \ge 1$) by considering the function $ \displaystyle f(z) = \frac{\pi \csc \pi z}{z^{4n-1} \sinh \pi z}$ ...
5
votes
0answers
100 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 ...
5
votes
0answers
163 views

An entire function of strict order 2

Here is a problem from Stein and Shakarchi Complex Analysis, can somebody help me to solve it? I guess we can use Phragmen-Lindelof theorem but I don't know the exact way. Suppose $f(z)$ is an entire ...
5
votes
0answers
54 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
189 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 ...
5
votes
0answers
120 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
142 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 ...
5
votes
0answers
74 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
127 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
116 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 ...
5
votes
0answers
173 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
237 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
1k 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
123 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
92 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
335 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
763 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 ...
5
votes
0answers
296 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} = ...
4
votes
0answers
75 views

Ugly-nice double series

I'm trying to evaluate the following ugly double sum (presented in raw notation as used in my calculations): $\sum _{m=1}^{\infty } \sum _{n=1}^{\infty } \frac{4 m \cos \left(\frac{2 \pi m ...
4
votes
0answers
47 views

enitre function that preserve the rationals?

Here's a question i would be curious to know the answer The question is: what is the set of all entire functions $f: \mathbb{C} \to \mathbb{C}$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$.
4
votes
0answers
86 views

Prove that a series is $O(t^a)$.

Consider the series $$ u(t,x) = \sum_{i \geq 1} {u_i(x) t_1^i } + \sum_{i+2j \geq k+2, j\geq 1} {\varphi_{i,j,k}(x) t_1^i t_2^j y^k} $$ where $t \in \tilde{\mathbb{C} \setminus \{ 0 \}}$, $x$ is ...
4
votes
0answers
54 views

Can we use a sum of residues to develop an asymptotic expansion for this unknown function?

In the course of solving a particular physical problem, I have derived a relationship between two unknown functions: $$ f(s) = \frac{s \sinh{\frac{\pi s}{2}}}{2 \pi i \beta} \int_{-c- i ...
4
votes
0answers
42 views

Soft Question: Inequalities like this

I am studying signed and complex measure and at a point in a proof the following lemma is being used: Lemma. If $z_1,...,z_n$ are complex numbers, then there exists a subset $S\subset\{1,2,...,n\}$ ...
4
votes
0answers
75 views

Showing that $\lim_{N \to \infty} \int_{|z|=N+\frac{1}{2}} \frac{ \sinh az}{\sinh \pi z} \mathrm{e}^{ibz} \ dz =0$

To evaluate $ \displaystyle \int_{0}^{\infty} \frac{\sinh ax}{\sinh \pi x} \cos (bx) \ dx \ (a< \pi)$, you could let $ \displaystyle f(z) = \frac{\mathrm{e}^{(a+ib)z}}{\sinh \pi z} $ and integrate ...
4
votes
0answers
84 views

Determining “good” contours for evaluating integrals

This is more of a general question, but I'll lead with an example. Suppose we wish to evaluate $\displaystyle \int_0^{\infty} \dfrac{1}{1+x^7} dx$ The goal, it seems, is to find nice contours which ...
4
votes
0answers
57 views

Probably Riemann surface integral

Here is the integral: May you please suggest some beautiful idea on using Riemann surface, or some Gauss-Ostrogradsky at the beginning. Also, the initial integral looks really symmetric, so maybe ...
4
votes
0answers
88 views

Number of zeros equal number of linearly independent analytic functions

I'm trying to read this paper and I'm stuck on a particular point. The authors are constructing an analytic function $f(z)$ which have to satisfy the following boundary conditions: ...
4
votes
0answers
55 views

Verma modules and delta function

What is the relationship between Verma modules and delta functions? Here's the quote from Woit's notes on Lie theory (http://www.math.columbia.edu/~woit/LieGroups-2012/vermamodules.pdf): The ...
4
votes
0answers
36 views

Continuous complex functions.

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq g(\partial ...
4
votes
0answers
213 views

Find a conformal map from the disc to the first quadrant.

Find a conformal mapping of the disk $x^2+(y-1)^2\lt 1$ onto the first quadrant $x, y \gt 0$ I did something, which may be false or not, I cannot exactly say anything. I used the composition of ...
4
votes
0answers
145 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
4
votes
0answers
62 views

Is there a way to calculate $\int \limits_0^1\frac{x^3}{\sqrt{x^2-1}}\frac{1}{1-a^2x^2}\frac{1}{1-b^2x^2}\frac{1}{c-x}\mathrm dx$

I want to calculate $\displaystyle \int \limits_0^1\dfrac{x^3}{\sqrt{x^2-1}}\dfrac{1}{1-a^2x^2}\dfrac{1}{1-b^2x^2}\dfrac{1}{c-x}\mathrm dx$ $a$ and $b$ are real parameters, c could be complex and is ...