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Showing posts with label Complex Analysis. Show all posts
Showing posts with label Complex Analysis. Show all posts

Monday, December 5, 2016

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

Monday, October 3, 2016

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

Wednesday, March 16, 2016

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

Sunday, February 7, 2016

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

Friday, December 25, 2015

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

Thursday, December 17, 2015

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

Wednesday, October 21, 2015

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

Thursday, August 20, 2015

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

Friday, April 10, 2015

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: