Search found 597 matches

by Tolaso J Kos
Mon Nov 09, 2015 1:04 pm
Forum: General Topology
Topic: Rendezvous value
Replies: 3
Views: 3532

Rendezvous value

Let \( (X, d) \) be a complete and a connected metric space. Prove that there exists a unique number \( r=r(X, d)>0 \) with the property: For all \( n \in \mathbb{N} \) and for all \( x_i, \; i=1,2,\dots, n \) there exists \( z \in X \) such that \( \displaystyle \frac{1}{n} \sum_{i=1}^{n} d(z, x_i)...
by Tolaso J Kos
Mon Nov 09, 2015 1:00 pm
Forum: Complex Analysis
Topic: The function $f$ is constant
Replies: 2
Views: 4040

The function $f$ is constant

Let \( f \) be an entire function across the complex plane. If \( \mathfrak{Im}(f(z))>\mathfrak{Re}^2 (f(z))-2 \) holds, then prove that \( f \) is constant.
by Tolaso J Kos
Mon Nov 09, 2015 12:48 pm
Forum: Real Analysis
Topic: Fourier series and a known identity
Replies: 1
Views: 2180

Fourier series and a known identity

Let \( f(x) =e^{ax} , \;\; x \in [-\pi, \pi)\) . Show that the Fourier series of \( f \) converges in \( [-\pi, \pi) \) to \( f \) and at \( x = \pi \) to \( \displaystyle \frac{e^{a\pi}+e^{-a\pi}}{2} \). Deduce that:


$$\frac{a\pi}{\tanh a\pi}=1+\sum_{n=1}^{\infty}\frac{2a^2}{n^2+a^2}$$
by Tolaso J Kos
Mon Nov 09, 2015 12:44 pm
Forum: General Mathematics
Topic: Monotony of a function
Replies: 0
Views: 1918

Monotony of a function

Examine the monotony of the function:

$$f(j)=\prod_{i=-j}^{0}\sum_{k=0}^{\infty}\frac{i^k}{k!}, \; j \in \mathbb{Z}$$
by Tolaso J Kos
Mon Nov 09, 2015 12:40 pm
Forum: General Topology
Topic: Compact Polish Space
Replies: 1
Views: 2516

Compact Polish Space

Let \( X = [0,+\infty)\cup\{+\infty\} \). Endow it with the metric

$$\rho(x, y)=|\arctan x - \arctan y |$$

Prove that \( X \) under \( \rho \) is separable, complete and compact.
by Tolaso J Kos
Mon Nov 09, 2015 12:36 pm
Forum: Complex Analysis
Topic: Rational function
Replies: 0
Views: 1927

Rational function

Let

$$R(z)= \sum \frac{1}{\log^2 z}, \; z \in \mathbb{C} \setminus \left \{ 0, 1 \right \}$$

where the summation is taken over all branches of the logarithm. Prove that $R$ is a rational function and deduce its formula.
Source
I have taken this exercise from the American Mathematical Monthly.
by Tolaso J Kos
Mon Nov 09, 2015 12:22 pm
Forum: Real Analysis
Topic: Intersection non empty
Replies: 0
Views: 1920

Intersection non empty

Let $\{V_n \}$ be a nested , decreasing sequence of open sets; each of which contains a finite union of intervals whose total length is above some fixed bound $\epsilon$. Prove that $\displaystyle \bigcap_{n=1}^{\infty} V_n \neq \varnothing$.