Articles of complex analysis

Prove that a complex valued polynomial over two variables has infinitely many zeroes

This is a homework question that I am struggling with. Given a polynomial over the complex numbers in two variables, show that the polynomial has infinitely many zeroes. So let’s say that the polynomial is a functions of $u$ and $v$. Let’s consider the polynomial as a function of $u$, with $v$ as a parameter: […]

Show that for any $w \in \mathbb{C}$ there exists a sequence $z_n$ s.t. $f(z_n) \rightarrow w$

I want to prove the following: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be holomorphic and non-constant. Then for $w \in \mathbb{C}$ there exists a sequence $(z_n)_{n \in \mathbb{N}} \subset \mathbb{C}$ with $lim_{n\rightarrow w}f(z_n) = w$. Which theorem can I use here? I know that by Liouville $f$ must be unbounded but does that help me? Can […]

A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$

Yesterday, during reviewing my old lecture notes on advanced quantum mechanics, i stumbeled over the following integral identity, which seems, on a first glance, too nice to be true $$ I_{A,B}=\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy=\sqrt{\frac{i\pi}{B }}e^{i(\sqrt{A}+\sqrt{B})^2} $$ with $A,B>0$ After working on it for a few hours i came up with a solution, which i think is […]

Continuous extension of a Bounded Holomorphic Function on $\mathbb{C}\setminus K$

Let $f:\mathbb{C}\setminus K\rightarrow\mathbb{D}$ be a holomorphic map, where $K$ is a compact set with empty interior. My question: Prove or disprove that: $f$ extends continuously on $\mathbb{C}.$ Remark: Observe that if $K$ is discrete then by the Riemann Removable Singularity Theorem we know that infact there is a holomorphic extension.

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 a conformal map, which is conformal. Firstly, let’s get a conformal map from the disk to the unit disk, […]

$f^2+2f+1$ is a polynomial implies that $f$ is a polynomial

This is a complex analysis problem. Let $f$ be an entire function and $f^2+2f+1$ be a polynomial. Prove that $f$ is a polynomial.

Showing that $\int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} = \pi\sqrt{2}$

Show that $$ \int_{-\pi}^{\pi} \frac{d\theta}{1+\sin^2(\theta)} = \pi\sqrt{2} $$

Convergence of Roots for an analytic function

Show that the roots of $$ f(z) = z^n+z^3+z+2 =0 $$ converge to the circle $|z|=1$ as $n \to \infty$.

Prove that $|\sin z| \geq |\sin x|$ and $|\cos z| \geq |\cos x|$

For any $z=x+iy$, prove the following: $$|\sin z| \geq |\sin x|$$ $$|\cos z| \geq |\cos x|$$ $\epsilon$-$\delta $ proof is not required. I don’t really know how to proceed. I know in order to remove the absolute values I can square both sides and I have tried proving this statement using the hyperbolic forms and […]

Is there a Möbius transformation that scales disks to the unit disk?

When working in the complex plane, often times I would like to scale a disk $|z-z_0|<R$ to the unit disk. I would first translate $z_0$ to the origin, but after that, what can we multiply by to scale the radius down from $R$ to $1$? I’m curious because in reading a proof of Schwarz’ lemma, […]