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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A problem involves Mittag-Leffler theorem and Weierstass Theorem

$G$ is a region,$\{a_n\}$ and $\{b_m\}$ are two sequence of distinct points in $G$ without limit in G.And $a_n\neq b_m$ for all $n,m$.Let $S_n(z)=\sum_{j=1}^{m_n}\frac{A_{jn}}{(z-a_n)^j} $.We are ...
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233 views

Partial fraction development of $\cot \pi z$

"Compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ by comparison to the partial fraction development of $\cot \pi z$." I'm not sure what "the partial ...
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72 views

How can I transform the fuction $x=y$ under the complex function $f(z)=z^2+z$?

How can I transform the fuction $x=y$ under the complex function $f(z)=z^2+z$??? First I define the horizontals and verticals lines of the complex plane. Then if $z=x+iy$ the function ...
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96 views

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
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43 views

Do there exist two distinct such points?

Please tell me if my attempt on the following problem will work: Suppose $f(z)$ is analytic in $|z|≤r.$ Do there exist two distinct points $z_0 = re^{iθ_0}$ and $z_1 = re^{iθ_1}$ such that ...
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136 views

Integrating the exponential of a complex quadratic matrix

Problem statement I'm trying to do a discretized path integral/functional integral. The integral that I'm stuck with is of the form $$ \int_{-\infty}^{+\infty} \mathrm{d}^3\vec{x}_1\, ...
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514 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 ...
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74 views

$\mathrm{e}^{-1/x^2}$ real differentiable but not complex differentiable at $x=0$

So I was reading about Laurent series and $\mathrm{e}^{-1/x^2}$ was used as an example. We define the function $f(x) = \mathrm{e}^{-1/x^2}$ for $x \neq 0$ and $f(0)=0$. Then it was stated that, as a ...
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118 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
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150 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
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56 views

Analytic Continuation of a nowhere existing Mellin Transform

I'm trying to give sense to the (self-made) statement: The analytic continuation of $\int_0^\infty e^t\;t^{s-1}\;dt$ is holomorphic in $s=0$. At first sight this could seem completely ...
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515 views

What is the principal branch of $f(z)=\sqrt{1-z}$, $z\in\mathbb{C}$?

My question is, essencially, about the definition of the principal branch of a function that is not a Logarithm. In this case, $f(z)=\sqrt{1-z}$.
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69 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
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277 views

Separate incomplete elliptic integral into real and imaginary parts

I am working in a problem that involves Incomplete Elliptic Integrals of the First and Second kind of the form $F(\sin^{-1}x~|~m)$ and $E(\sin^{-1}x~|~m)$ where the parameters $m$, $x$ are real ...
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372 views

Set of derivatives of a normal family of analytic functions is itself a normal family.

I'm working on a problem that is easily solvable if I can prove the statement in the title. Here's what I've done so far: Given $\mathscr{F}$ a normal family of analytic functions, let ...
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80 views

Is a complex polynomial a regular covering? What is its group of deck tranformations?

We know that a complex polynomial $P$ of degree $n$ is an $n$-sheeted covering from $$\{\mathbb{C} - P^{-1}\{\text{critical values of }P\}\} \to \{\mathbb{C} - \{\text{critical values of }P\}\}. $$ ...
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242 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
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84 views

Show that $\int_{\alpha}\frac{1}{z}\, dz=\int_{\beta}\frac{1}{z}\, dz$.

Let $a$ and $b$ be positive real numbers. Define ways $\alpha,\beta\colon [0,1]\to\mathbb{C}$ via $$ \alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t),~~~~~\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t). $$ Show ...
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154 views

Is $\sum_{n=1}^{\infty} \frac{z^2}{z^2 n^2 +1}$ a meromorphic function?

I came across this old exam problem that I think it is a typo but I want to make sure there is not a problem with my knowledge. The problem is show $$\sum_{n=1}^{\infty} \frac{z^2}{z^2 n^2 +1}$$ ...
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267 views

Complex Analysis and Proper Holomorphic Maps

Prove there is no proper holomorphic map from the unit disc into the complex plane. I know as $z$ approaches the boundary of the unit disc, $f(z)$ will approach infinity, hence unbounded. Can we ...
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45 views

Infinite differential equations

Let $r\in\mathbb N$ and $f$ be an entire function on $\mathbb C$ such that for every $R\in\mathbb C[z]$, there exist polynomials $P_{i,R}(z)\in\mathbb{C}[z]$ ($0\le i\le r$) not all zero such that, ...
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155 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
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326 views

If the product of two analytic functions is zero, then one must be identically zero.

I want to prove this statement: Let $f,g$ be analytic on $D(0,2)$. If $f(z)g(z) = 0$ when $z = 1/n$ for $n \in \mathbb{N}$, then either $f \equiv 0$ or $g \equiv 0$ in $D(0,2)$. My attempt: ...
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160 views

Series expansions of branches of multivalued functions

In David Wunsch's Complex Variables with Applications, Example 3 on page 266 asks the reader to find a Maclaurin expansion of $f(x)=(z+1)^{1/2}$ where the principal branch is used. The principal ...
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162 views

Integral form of $2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$

Being inspired by this post, I've wondered if the infinite series below may be expressed as an intregral. I'm very curious about that. $$2\sum_{k=1}^{\infty}\frac{(2k-1)^2-1}{(2k-1)^4+(2k-1)^2+1}$$ ...
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329 views

Finding a sequence of polynomials that converges uniformly to a holomorphic function on an open set

The following is exercise 13.2 in Rudin's Real & Complex Analysis, which I'm self-studying. Let $\Omega = \{z: |z| < 1 \text{ and } |2z - 1| > 1\}$, and suppose $f \in H(\Omega)$. Must ...
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313 views

Zeros of analytic function and limit points at boundary

Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose ...
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164 views

Residue at an essential singularity

Consider the function $$f(z)=\frac{e^{\frac{1}{z-1}}}{e^z -1}$$ $z_0=1$ is an essential singularity, hence $$f(z)=\displaystyle\sum_{-\infty}^{+\infty}a_n(z-1)^n$$ near to $z_0=1$ and I want to find ...
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335 views

Under which hypotheses is switching between sum and integral signs legit?

Which hypotheses are needed to change the order of sum and integral signs? Concrete example: consider the expression $$ ...
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93 views

Integration of sine^2 w.r.t. some norm

Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$. How much ...
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54 views

Holomorphic analogue of geodesics

Let $X$ be a complex manifold with a Hermitian metric. Is there a "complex" analogue of geodesics on $X$ which is of any interest? For example, is anything known about holomorphic maps $f : \mathbb C ...
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126 views

Finding the analytic function

Find all analytic function $f: \mathbb C \rightarrow \mathbb C$ such that $|f^`(z)|$ constant on curves of the form $Ref$ constant. This is one of the past comp question. Seriously I do not know ...
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79 views

$f(r)\in\mathbb R$ $\forall$ real $r<-1$

$f$ is analytic in {$z:|z|>1$} and $f(r)\in\mathbb R$ $\forall$ real $r>1$. How can I show that the same hold $\forall$ real $r<-1$? Please don't solve it completely. I'm just looking ...
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62 views

Perturbations of algebraic varieties

Let $P(z,w):\mathbb C^2\to\mathbb C$ be a certain polynomial, and consider $p(s,t)=P(e^{is},e^{it})$ its restriction to the torus. In the specific problem I'm considering, the set $Z=\{(s,t): ...
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340 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 ...
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181 views

Proveing that if a Holomorphic function is one to one on a circle, then it's one to one within the disk created by the circle.

I would like to prove that if a function $f(z)$ is holomorphic on $\overline{D(P,r)}$ and one to one on $\partial D(P,r)$ then $f$ is one to one on $D(P,r)$. I noticed that for $w \in f(D(P,r))$ with ...
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75 views

Complex nonlinear differential equation

I have the following nonlinear differential equation: $$\ddot z(t)-\sin(z(t))=0$$ where $z(t)$ is a complex variable. The solution of the same equation with $z(t)$ real, is a function of Jacobi ...
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799 views

Uniform distribution on the unit circle (in the complex plane)

I was trying to prove that for a standard complex Gaussian variable $Z$ it holds that $|Z|^2$ is exponentially distributed with parameter 1, $\frac{Z}{|Z|}$ is uniformly distributed on the unit circle ...
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595 views

schwarz lemma, a simple example

Let $f$ be a holomorphic function on $B(0,R)$ , R>0. Assume that there exist an $M>0$ such that $| f(z) | \le M$ $\forall z\in B(0,R)$. , and a natural number, such that : $$ 0 = f(0)=f'(0) = ... ...
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73 views

Unramified functions between Riemann surfaces

Let $F:X\rightarrow Y$ a unramified holomorphic function between two compact Riemann surfaces. I don't understand why $F$ is a covering map. By a well-known theorem $F$ is surjective; then since the ...
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230 views

Anti-holomorphic involution of $\mathbb{P}^1$

I wonder if anti-holomorphic involution of $\mathbb{P}^1$ is, up to change of coordinate, given by either $$ z\mapsto \overline{z}, \ \ \,z\mapsto -\overline{z}, \ \ \ or \ \ \ z\mapsto ...
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753 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 ...
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267 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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382 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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405 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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208 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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169 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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355 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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146 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 ...
3
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220 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 ...