Search found 597 matches
- Mon Nov 09, 2015 1:04 pm
- Forum: General Topology
- Topic: Rendezvous value
- Replies: 3
- Views: 3610
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)...
- Mon Nov 09, 2015 1:00 pm
- Forum: Complex Analysis
- Topic: The function $f$ is constant
- Replies: 2
- Views: 4605
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.
- Mon Nov 09, 2015 12:48 pm
- Forum: Real Analysis
- Topic: Fourier series and a known identity
- Replies: 1
- Views: 2200
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:
$$\frac{a\pi}{\tanh a\pi}=1+\sum_{n=1}^{\infty}\frac{2a^2}{n^2+a^2}$$
$$\frac{a\pi}{\tanh a\pi}=1+\sum_{n=1}^{\infty}\frac{2a^2}{n^2+a^2}$$
- Mon Nov 09, 2015 12:44 pm
- Forum: General Mathematics
- Topic: Monotony of a function
- Replies: 0
- Views: 1966
Monotony of a function
Examine the monotony of the function:
$$f(j)=\prod_{i=-j}^{0}\sum_{k=0}^{\infty}\frac{i^k}{k!}, \; j \in \mathbb{Z}$$
$$f(j)=\prod_{i=-j}^{0}\sum_{k=0}^{\infty}\frac{i^k}{k!}, \; j \in \mathbb{Z}$$
- Mon Nov 09, 2015 12:40 pm
- Forum: General Topology
- Topic: Compact Polish Space
- Replies: 1
- Views: 2578
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.
$$\rho(x, y)=|\arctan x - \arctan y |$$
Prove that \( X \) under \( \rho \) is separable, complete and compact.
- Mon Nov 09, 2015 12:36 pm
- Forum: Complex Analysis
- Topic: Rational function
- Replies: 0
- Views: 1986
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.
$$R(z)= \sum \frac{1}{\log^2 z}, \; z \in \mathbb{C} \setminus \left \{ 0, 1 \right \}$$
where the summation is taken over all branches of the logarithm. Prove that $R$ is a rational function and deduce its formula.
Source
- Mon Nov 09, 2015 12:22 pm
- Forum: Real Analysis
- Topic: Intersection non empty
- Replies: 0
- Views: 1989
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 $\displaystyle \bigcap_{n=1}^{\infty} V_n \neq \varnothing$.