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)

5
votes
0answers
47 views

Angle between two curves at infinity?

In complex analysis, how would I find the angle between the two curves $f(z)=0$ and $g(z)=0$ at $\infty$? My guess would be that we would let $w=\frac{1}{z}$ and find the angle between the two ...
5
votes
0answers
69 views

Finding an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$

I encountered the following problem in the lecture note in my complex analysis class: Problem. Find an explicit entire function $g$ satisfying $g(n \log n) = n^{\pi}$ for $n = 1, 2, \cdots$. ...
5
votes
0answers
65 views

Geometry of the zeros of a power series.

This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series $$ f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, ...
5
votes
0answers
147 views

Relation between hypergeometric and gamma functions

Show that, for a positive integer $n$, ...
5
votes
0answers
44 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 $z, w \in U$, ...
5
votes
0answers
78 views

Integrate $\int x|J_1(x)|^2 \, dx$

Hi I am trying to integrate: $$ \int_0^R x|J_1(bx)|^2 \, dx,\quad R>0, b\in \mathbb{C} $$ where $|J_1|^2=\bar{J}_1 J_1$ where $\bar{J_1}$ means complex conjugation of the bessel function. My ...
5
votes
0answers
59 views

Weierstrass Product of $z\cos(z)-\sin(z)$

On a black board in the floor of my university was this question as a puzzle, and I would like to know why the $\exp(g(z))$ in the Weierstrass Product is constant $1$, in the question one answer ...
5
votes
0answers
46 views

Use of residues on a linear contour

As one step in Ahlfors derivation of Stirling's formula (2nd edition pages 199-200), he uses residues to evaluate the integral $$\int\limits_{\xi = n+1/2} \frac{d\zeta}{\left|\zeta +z\right|^2}$$ ...
5
votes
0answers
59 views

Generalized Trigonometric Functions in terms of exponentials and roots of unity

I am trying to come up with generalized trigonometric functions using the exponential definition that we use today for the trig functions sine and cosine $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}; \cos x ...
5
votes
0answers
47 views

Radial limits of composition of functions

Is it true that if $f\in H(U)$ is a holomorphic function whose nontangential limits exist a.e and $g\in H^\infty(U)$ is a nonconstant holomorphic function whose range is in $U$ and whose radial limits ...
5
votes
0answers
88 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
73 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
209 views

Analytic variety is a countable union of complex manifolds

In an article on real analytic manifolds I came across the following remark: Let $W$ be a purely $k$-dimensional analytic subvariety of a domain in $\mathbb{C}^n$ and let $S$ be its singular locus. ...
5
votes
0answers
74 views

A periodic entire function which must have a fixed point

I would like to check my work on the following problem: Suppose $f(z)$ is a non-constant periodic entire function satisfying $f(z+1)=f(z)$. Show that $f(z)$ has a fixed point. So my attempt is: ...
5
votes
0answers
67 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
163 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
299 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.
5
votes
0answers
67 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
131 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
165 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
203 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
80 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
216 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) = ...
5
votes
0answers
194 views

How do zeros on the complex plane affect the real number line?

Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
5
votes
0answers
80 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
184 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
153 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
504 views

Question Relating with Open Mapping Theorem for Analytic Functions

This problem is taken from Section VIII.4 of Theodore Gamelin's Complex Analysis: Let $f(z)$ be an analytic function on the open unit disk $\mathbb{D}=\{|z|<1\}$. Suppose there is an annulus ...
5
votes
0answers
754 views

Solve $\int \cos^{2n}\theta d\theta$

I am trying to solve the integral $\int_0^{2\pi} \cos^{2n}\theta d\theta$ using residues. I get the wrong answer so could you please say what I am doing wrong? We start with the substitution $z = ...
5
votes
0answers
188 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
106 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
458 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
361 views

Visualizing the domain of the square root

I would like to show someone the domain of the complex square root function (the 2-sheeted riemann surface). Is there a good interactive visualization software for this? I would like some sort of ...
5
votes
0answers
498 views

An obscure explanation in Conway's Complex analysis

I don't understand a paragraph in Conway's complex analysis at the beginning of Chapter VI page 128 (Maximum Modulus Theorem). He says: "Note that in Theorem 1.2 we did not assume that $G$ is ...
5
votes
0answers
174 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?
5
votes
0answers
1k 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
363 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
44 views

Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq ...
4
votes
0answers
36 views

Is there an analytic function $f$ in the open unit disc such that $|f(z)|=e^{|z|}$ therein?

Is there an analytic function $f$ in the open unit disc such that $|f(z)|=e^{|z|}$ therein? My try: Suppose such a function exists .Then $|f(0)|=e^0=1$ Also $|f|\ge 0$. Now $e^{|z|}$ attains its ...
4
votes
0answers
62 views

Holomorphic function on a connected set that has 1 fixed point

I am currently studying complex analysis, and I came across an exercice that I could only partially solve, here it is : Let $\Omega$ be a connected bounded set of $\mathbb{C}$, let $f : \Omega ...
4
votes
0answers
82 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
4
votes
0answers
104 views

What is THE domain of analyticity of a holomorphic function?

I am self-studying complex analysis, and I am a little bit confused on notations. Suppose that $f:U \to \mathbb C $ is a holomorphic function defined on an open subset of $\mathbb C^n $. I understand ...
4
votes
0answers
31 views

Tensor product of $L^1([0,1],\omega)$ with some Banach space

We know that for any Banach space $E$ if $\hat{\otimes}$ denotes projective tensor product then $L^1([0,1])\hat{\otimes} E$ is isomorphism isometric with $$L^1([0,1],E)=\{f:[0,1]\to E:\ \ ...
4
votes
0answers
83 views

Double contour integral in terms of real integrals

Let $\gamma$ be a curve in $\mathbb{C}$, and let $\gamma_0$ be a circle in an open connected set $A \subset \mathbb{C}$ around $z_0 \in A$. Suppose the interior of $\gamma_0$ lies in $A$. Let $z$ be ...
4
votes
0answers
97 views

Trigonometric Expression for $1 + \cos \alpha + \cos 2\alpha + \cdots + \cos n \alpha$ using complex numbers

This question is not a duplicate because I am asked here to use the fact that $1 + \cos \alpha + \cos 2 \alpha + \cdots + \cos n \alpha = Re (1 + z + z^{2} + \cdots + z^{n})$, where the question this ...
4
votes
0answers
53 views

Find the value of $a_{-2}$ in the Laurent series expansion of $\sin{(\frac{z}{z+1})}$

If $\sum_{-\infty}^{\infty}a_n (z+1)^n$ is the Laurent series expansion of $f(z)=\sin{(\frac{z}{z+1})}$, then find the value of $a_{-2}$. My work: So we are asked to find the laurent series ...
4
votes
0answers
67 views

inverse Laplace transform by finding residues of essential singularities

I want to find the inverse Laplace transform of $$F(s)=\exp\Big(-\sqrt{2s}\tanh(\sqrt{2s})\Big).$$ Despite the square roots, $F$ doesn't have any branch points since ...
4
votes
0answers
51 views

Double Holomorphic functions

Suppose I have two complex bivariate homolorphic functions $f(z,w)$ and $g(z,w$) such that $$ \frac{\partial f}{\partial z} = \frac{\partial g}{\partial w} $$ $$ \frac{\partial f}{\partial w} = - ...
4
votes
0answers
65 views

Show that the “Hartogs Regularity Radius” $R(z)$ is subharmonic

Exercise I'm a little stuck on an Exercise in Holomorphic Functions and Integral Representations in Several Complex Variables by R. Michael Range. The Exercise (E.II.5.1) is as follows (here ...
4
votes
0answers
152 views

Find the complex (or real) roots of $e^{\frac{3 x}{2}}+2 \cos \left(\frac{\sqrt{3} x}{2}\right)$

Define for natural $n\geq 2$ $$G(x,n)= \sum _{k=0}^\infty \frac{x^{k n}}{(k n)!}= \frac{\sum _{k=0}^{n-1} e^{x e^{\frac{2 i \pi k}{n}}}}{n}= G(x e^{\frac{2 i \pi}{n}},n)= \prod_{m=1}^\infty ...