The complex sequence does not converge uniformly

Prove that there does not exist a sequence $\{ p_n(z)\}_{n \in \mathbb{N}}$ of complex polynomials such that $p_n(z) \rightarrow \frac{1}{z}$ uniformly on $\mathcal{C}_R=\{ z \in \mathbb{C} \mid \left| z \right| = R\}$.

Solution

The funtion is identically zero

Let $f$ be an entire function such that

$$\left| f(z) \right| \leq \log \left( 1 + \left| z \right| \right) \quad \text{forall} \; z \in \mathbb{C}$$

Show that $f(z)=0 $ forall $z \in \mathbb{C}$.

Solution

On an entire function

Let $n \in \mathbb{N}$ and let $f$ be an entire function. Prove that for any arbitrary positive numbers $a, b$ it holds that:

$$\frac{\bigintsss_{0}^{2\pi} e^{-i n t}f \left ( z+a e^{it} \right ) \, {\rm d}t}{\bigintsss_{0}^{2\pi} e^{-i n t} f\left ( z + b e^{it} \right ) \, {\rm d}t} = \left ( \frac{a}{b} \right )^n$$

Solution

On zeros and poles of a meromorphic function

Let \( f \) be a meromorphic function on a (connected) Riemann Surface \( X \). Show that the zeros and the poles of \( f \) are isolated points.

Solution

Meromorphic functions on the Riemann sphere

Show that every meromorphic function \( f \) on the Riemann sphere \( \hat{\mathbb{C}} \) is rational, i.e. of the form \( p \over q \), where \( p \) and \( q \) are coprime polynomials.

Solution

Holomorphic functions

Let $\mathbb{D}= \{z \in \mathbb{C}: |z|<1 \}$ be the complex unit disk and let $0<a<1$ be a real number. Suppose that $f:\mathbb{D} \rightarrow \mathbb{C}$ is a holomorphic function such that $f(a)=1$ and $f(-a)=-1$.

a) Prove that $\sup \limits_{z \in \mathbb{D}} \left| f(z) \right| \geq \frac{1}{a}$.
b) Prove that if $f$ has no root , then:

$$\sup_{z \in \mathbb{D}} \left|f(z)\right|\geq \exp \left( \frac{1-a^2}{4a} \pi \right)$$

(Schweitzer Competition, 2012)

Solution

A relation with complex unity and logs.

Let $n \in \mathbb{N}$ then prove that:

$$\int_0^{\pi/2} (\log \cos x+ \log 2 + ix)^n \, {\rm d}x + \int_0^{\pi/2} (\log \cos x + \log 2 - ix)^n \, {\rm d}x=0$$

Solution

Lusin Areal Integral Formula

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be an analytic and $1-1$ function and let $\mathbb{D}$ be the open unitary disk. Prove that:

$$\iint \limits_{\mathbb{D}} \left |f'(z) \right| \, {\rm d}z = f \left(\mathbb{D} \right)$$

Solution

Contour integral

Let $a,  b \in \mathbb{C}$ and $|b|<1$. Evaluate the contour integral:

$$\frac{1}{2\pi} \oint \limits_{|z|=1} \left| \frac{z-a}{z-b}\right|^2 \;|{\rm d}z| = \frac{|a-b|^2}{1-|b|^2}+1$$

Solution

Contour Integral

Consider the branch of $ f(z) =\sqrt{z^2-1} $ which is defined outside the segment $ [-1, 1] $ and which coincides with the positive square root $ \sqrt{x^2-1} $ for $ x>1 $. Let $ R>1 $ then evaluate the contour integral:

$$\oint_{\left |  z\right |=R}\frac{{\rm d}z}{\sqrt{z^2-1}}$$

Solution: