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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556 views

Integration of nontrivial trigonometric functions

First an example which I know how to solve. If we have the following integral $$\int_{-\pi}^{\pi}\frac{1}{1+3~\cos^2(t)}dt$$ there is a very practical way to evaluate it by interpreting it as some ...
3
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222 views

Properties of the lemniscate functions as meromorphic functions on $\mathbb{C}$

We consider the following function. $$u(x) = \int_{0}^{x} \frac{dt}{\sqrt{1 - t^4}}$$ $u(x)$ is defined on $[-1, 1]$. Since $u'(x) = \frac{1}{\sqrt{1 - x^4}} > 0$ on $(-1, 1)$, $u(x)$ is strctly ...
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338 views

Family of One-to-One Analytic Functions is Normal

I've been working on the following problem: Let $D$ and $D'$ be simply connected plane domains, each different from the whole plane. Suppose that $z_1 \in D$ and $z_2 \in D'$. Let $\mathcal{F}$ be ...
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356 views

Uniform convergence of analytic functions

Let $f_{n}(z), g(z)$ be entire functions, for all $n\geq 1$. Suppose that $g(x)$ doesn't vanish on $\mathbb H\cup\mathbb R$ (so we have $\frac{f_{n}(z)}{g(z)}$ analytic on $\mathbb H\cup\mathbb R$). ...
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197 views

complex analysis question

Prove that if $f$ is an analytic function in an open set containing the closed unit disk and if $k$ is a positive integer, then there exists a $z_0$ with $|z_0|=1$ and ...
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154 views

Equivalent Definitions of the Weierstass $\wp$-Function

I've come across two equivalent definitions of the Weierstrass $\wp$-function, but don't know how to prove that they are equivalent. Definition 1 $\wp(z)=cf(z)+d$ where $f$ is the elliptic function ...
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297 views

Jensen's Inequality for complex functions

Jensen's inequality states that if $\mu$ is a probability measure on $X$, $\phi$ is convex, and $f$ is a real-valued function, then $$ \int \phi(f) \, d\mu \geq \phi\left(\int f \, d\mu\right).$$ Is ...
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144 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace ...
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192 views

Function in the Hardy space $\mathbb H^{2}$

Given a function $f(z)$, $z=x+iy, x,y\in \mathbb R$, which belongs to $\mathbb H^{2}(\mathbb C^{+})$, where $\mathbb C^{+}$ is the upper half plane Im$(z)>0$ and $f(a_{n})=0$, for all $n\in \mathbb ...
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227 views

Etymology of the word “pole”?

In his book Control System Design, Bernard Friedland writes (section 4.2, page 115): The roots of the denominator [of a rational function] are called the poles of the transfer function because ...
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104 views

Length of $\frac{\partial }{\partial z}$ in Kaehler geometry.

I am taking a Kaehler geometry course this semester. The book we use is Tian's Canonical Metrics in Kaehler Geometry. I got a little confused about the calculation there in. For example, ...
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146 views

Analytic function that provide $f^2(z)=z$

I am trying to solve this problem: Does there exist a function $f(z)$, that is analytic at $E=\{x+iy :x>y\}$ and provides $f^2(z)=z$ for every $z \in \mathbb C$. I have seen a solution that ...
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177 views

Constructing normalizations of algebraic curves vs constructing Riemann surfaces of functions

This question is sort of a further extension to this question I have been asking, Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface It seems ...
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72 views

Relation between n-tuple points on an algebaric curve and its pre-image in the normalizing Riemann surface

This question is sort of an extension to this previous question of mine, Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve If one knows the multiplicity of a ...
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277 views

Limit at infinity of a complex function

If $f(z)$ is an entire function such that $$ \lim_{x \rightarrow -\infty}\frac{f(x)}{|f(x)|}=1$$ where $|f(x)|$ is the modulus of $f$, and $f(x)$ is just evaluating $f$ at real $x$. What can we say ...
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82 views

Do the solutions to the unit equation lie dense in the complex numbers

Let $S\subset \overline{\mathbf{Q}}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of algebraic integers. ...
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102 views

Translating coordinates on a Riemann surface

Let $U\subset X$ be an open subset of a connected Riemann surface $X$. Let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. Let $P\in U$ be the ...
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102 views

Fermat-like equation

I'm looking for information on this kind of reverse-Fermat problem: Given $a,b,c \in \mathbb{C}$, find a $z \in \mathbb{C}$ such that $a^z + b^z = c^z$? When does such a $z$ exist? Is anything known? ...
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148 views

Describing a complex function

If $p,q,r\in \mathbb{C}$, how would one describe the curve $$\mathrm{Re}\left(pz^2+qz+r\right)=0.$$ If I write $p=p_1+ip_2$, $q=q_1+iq_2$, $r=r_1+ir_2$, and $z=x+iy$, then ...
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179 views

Change of variables in line integral with abs. value

Let $\gamma : I \rightarrow \mathbb C$ be a path. Let $g: \mathbb C \rightarrow \mathbb C$ be a biholomorphic map. Let $f$ be a holomorphic function. Consider the integral $$ \int_{g\circ \gamma} ...
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429 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 ...
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82 views

The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function

Let $X$ be a compact connected Riemann surface of genus $g>0$. We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
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522 views

Laplace inverse transform formula and Cauchy's integral formula

The question is about Laplace Transform and the inverse transform formula. Can the inverse transform formula be proved using Cauchy's integral formula?
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267 views

circle of convergence

How do you analyse for convergence/divergence on the circle of convergence of a particular power series?
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194 views

proving the saddle point method in a specific case

$1:$ the problem Let $f : U \to \Bbb{C}$ be analytic on some open set $U$ that includes the closed unit ball. Define a path $\gamma$ by: $\gamma(t) = e^{i t}$, -$\pi < t \leq \pi$ I want to ...
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226 views

Please recommend good text on complex Fourier series/analysis

I am looking for some good text/reference on complex Fourier series resp. Fourier analysis for complex (in particular holomoprhic) functions (of one variable). The more it contains on this particular ...
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29 views

Misplaced complex analysis intuition on Riemann Surfaces

Next week I will be giving a lecture, based on Chapter 2.6 from Jost's book Compact Riemann Surfaces. He states the following theorem: Theorem 1 (Jost Theorem 2.6.2) Let $S$ and $\Sigma$ be Riemann ...
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20 views

Contour Integral of sin(z)/(z^2-z)

Find the integral $\int_{\lambda}\frac{\sin(z)}{z(z-1)}$ where $\lambda(t) = 10e^{it},t\in[0,2\pi]$ We notice that there are poles at $z = 0$ and $z=1$. So we can use residue theorem but I am ...
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22 views

How to integrate $\int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}$?

In Lancaster & Blundell's QFT book they show that \begin{equation}A:= \int_{-\infty}^{\infty}dp \ p e^{ipx}e^{-it\sqrt{p^2+m^2}}\end{equation} returns a nonzero value for $x$, $t$ and $m$ ...
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54 views

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent

Find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{e^{nz^2}}{n}$ is convergent. I use a root test: $\lim_{n\rightarrow\infty} |\frac{e^{nz^2}}{n}|^{1/n}=\lim_{n\rightarrow\infty} ...
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55 views
+50

How prove exists $z_{0}$ such $|z_{0}|=1$,and $|f(z_{0})|\ge\frac{1}{2^{n-1}}\prod_{j=1}^{n}(1+|a_{j}|)$

let $a_{1},a_{2},\cdots,a_{n}\neq 0$ be given complex numbers and $$f(z)=\prod_{j=1}^{n}(z-a_{j})$$ I need to show that there exists a complex number $z_{0}$ such that $|z_{0}|=1$ and ...
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24 views

Solution of gaussian integral with hyperbolic cotangent

I was wondering if the integral $$I=\int_{-\infty}^{\infty}d\omega \omega e^{-(\omega/a)^2}\coth(\frac{b\omega}{2})\cos(\omega c)$$ where $a,b,c>0$ can be solved using complex countour ...
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90 views
+50

conformal mapping, regions of the complex plane marked +/-, find the function f,

The picture shows what the function f: $\mathbb{C}\to\mathbb{C}\cup\infty$ does to the plane. The values 0 at 0, 1 at $\pm$1, and $\infty$ at $\pm i$ are specified. To elaborate on the picture: ...
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14 views

Proof check: Continuity of an integral

$\gamma$ is a rectifiable curve in $\mathbb{C}$; For some open $G$, we have a continuous function $\phi:\{\gamma\}\times G\rightarrow \mathbb{C}$. $g(z):=\int_\gamma \phi(w,z) dw.$ We wish to show ...
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22 views

$f_j(z) $maps unit disk conformally

Suppose $f_j(z)\in H(D)$ maps the unit disk conformally to $\Omega_j , (j=1,2)$, where $\Omega_1\subset \Omega_2$ and $f_1(0)=f_2(0)$. Show that $$ |f'_1(0)|\leq |f'_2(0)|$$, and equality holds if and ...
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29 views

Harmonic function with boundary value 0 except one point

Let $\Delta=\{z\in\mathbb{C}: |z|<1\}$ Assume $u\in C(\overline{\Delta}\setminus \{1\})$ such that it is harmonic in $\Delta$ and $u(\xi)=0$ for $\xi\in S^1\setminus \{1\}$. (a) Find an example ...
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29 views

Laurent Series Expansion of $e^{z+\frac{1}{z}}$ about $z_0=0$

As the question title states, I'm tasked with finding the Laurent series expansion for $e^{z+\frac{1}{z}}$ about $z_0=0$. My approach is as follows. $$ \begin{align*} ...
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0answers
31 views

Image of $f(A)$ when $A$ is compact and $f$ is analytic in the interior of $A$ and continuous on the boundary of $A$

Suppose there is a non constant complex function $f$ and a compact set $A$ such that $f$ is analytic in the interior of $A$ and continuous on the boundary of $A$. Then clearly $f(A)$ is compact. Now ...
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19 views

Analytic function on unit disk - bounded derivative of origen

let $f(z)$ be an analytic function on unit disk $D=\{z:|z|<1\}$. $$d=\sup_{w,z\in D}|f(w)-f(z)|.$$ prove: $$|f'(0)|<\frac d 2.$$
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27 views

show a function is analytic on $\mathbb{C}$ by using power series

Use power series methods to show that the function which is $\frac{\sin z}{z} $ when $z \neq 0$ and $1$ when $z = 0$ is analytic on the whole complex plane. Any hint on how I might approach this? ...
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24 views

Necessary & Sufficient condition for the existence of Analytic Continuation

While solving the problems on Analytic continuation from Gamelin's book; I encountered this one- still unsolved: Let $D= \{0 < |z| < \epsilon\}$ and suppose $f$ is holomorphic at $z_{0} \in D$ ...
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24 views

Could characters in harmonic analysis be generalized into $S^2$?

Consider a locally compact abelian (LCA) group $G$. The start of commutative harmonic analysis is the fact that the collection of characters $\chi : G \to S^1$ (thought of as $S^1 = \mathbb{T} ...
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28 views

Asymptotics of inverse Laplace transform of a function with an essential singularity?

Let $h$ be the function $$ h(x) = \sum_{k\geq0} \frac{(ix)^k}{k!}\zeta(2k), $$ with the Laplace transform $$ \tilde h(s) = -\frac{\pi}{2s}\sqrt{i/s}\cot\left(\pi\sqrt{i/s}\right), $$ which has an ...
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34 views

Real integral with 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 ...
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43 views

Roots of $z^6+6z+10=0$

Find the number of roots of $z^6+6z+10=0$ in each quadrant. I want to use the argument principle. In the first quadrant, $f(z)=z^6+6z+10$ can be written as $f(R e^{i \theta})= R^6 e^{i6\theta}\{ ...
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31 views

How to relate two integration contour?

How one can relate two integration contour? For example if I have an integration contour like $\int_{-a}^{a}f(x)dx$ here let say a=infinity. How I can say that the integral $\int_{2}^{3}f(x)dx$ is a ...
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51 views

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$?

how to calculate $f(D(0,\delta) - \{ 0 \})$ with $f(z)=z\sin(\frac{1}{z})$ ?. I know that zero is an essential singularity, and so $f(D(0,\delta)-\{ 0 \})$ is dense in $\mathbb{C}$. This question ...
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0answers
23 views

contour intrgration, what's the right answer?

There exists an integral as follow: $$ \bar G(t)=\int_{-\infty}^{\infty}\frac{dE}{2\pi\hbar}e^{-iEt/\hbar}\frac{1}{E-\epsilon+i0^{+}} $$ My solution is: $$ {2\pi\hbar}\bar G(t)=-i\pi e^{-i\epsilon ...
2
votes
0answers
27 views

What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
2
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30 views

Showing that this is a biholomorphism.

I am supposed to prove that: $f(z)=\frac{-1}{2}(z+\frac{1}{z})$ is a biholomorphism from the upper unit disc: $\{z : |z|<1, \Im(z)>0\}$, to the upper half plane $\{z: > ...