Search found 284 matches

by Tsakanikas Nickos
Wed Jul 20, 2016 6:47 pm
Forum: Differential Geometry
Topic: On The Existence Of Shortest Paths
Replies: 4
Views: 5860

Re: On The Existence Of Shortest Paths

Let's prove some of the assertions then! Every compact metric space is boundedly compact (i.e. every closed and bounded subset is compact) : It is a fact (prove it if you want to) that every closed subset of a compact space is compact. In particular, any closed and bounded subset of a compact space ...
by Tsakanikas Nickos
Wed Jul 20, 2016 10:38 am
Forum: Differential Geometry
Topic: On The Existence Of Shortest Paths
Replies: 4
Views: 5860

Re: On The Existence Of Shortest Paths

$X$ is assumed to be boundedly compact. Can you justify the reduction to the case that $X$ is compact?


Furthermore, there is also a proof using Arzela-Ascoli's Theorem and the "semicontinuity of the lengths of curves".
by Tsakanikas Nickos
Tue Jul 19, 2016 9:37 pm
Forum: Differential Geometry
Topic: On The Existence Of Shortest Paths
Replies: 4
Views: 5860

On The Existence Of Shortest Paths

Let $(X,d)$ be a boundedly compact metric space. Show that whenever two points are joined by a rectifiable curve (i.e. a curve of finite length), then there exists a shortest path (minimal geodesic) between them.
by Tsakanikas Nickos
Fri Jul 15, 2016 6:31 pm
Forum: Calculus
Topic: Evaluation of an integral
Replies: 3
Views: 3293

Re: Evaluation of an integral

Thank you both for your nice solutions!
by Tsakanikas Nickos
Thu Jul 14, 2016 7:02 pm
Forum: Complex Analysis
Topic: Non existence of complex functions
Replies: 3
Views: 4586

Re: Non existence of complex functions

Show that a complex polynomial function \( \displaystyle P \) such that \( \displaystyle \Big| {e}^z + P(z) \Big| < e^{\Re(z)} \, , \, \forall z \in \mathbb{C} \) does not exist. Suppose that such a polynomial function \( \displaystyle P \) exists. Dividing \( \displaystyle \Big| {e}^z + P(z) \Big|...
by Tsakanikas Nickos
Thu Jul 14, 2016 7:00 pm
Forum: Complex Analysis
Topic: Non existence of complex functions
Replies: 3
Views: 4586

Non existence of complex functions

Show that the only entire function \( \displaystyle f \) that satisfies \( \displaystyle \Big| f(z)-i \Big| < \Big| f(z) \Big|e^{\Re(z)} \, , \, \forall z \in \mathbb{C} \) is the constant function \( \displaystyle f(z)=i \, , \, z \in \mathbb{C} \). Show that a complex polynomial function \( \disp...
by Tsakanikas Nickos
Thu Jul 14, 2016 6:58 pm
Forum: Complex Analysis
Topic: Evaluation of a limit
Replies: 0
Views: 2424

Evaluation of a limit

Evaluate the limit \[ \displaystyle \lim_{z \to 0} \frac{\mathrm{d}^2}{\mathrm{d}z^2} \left( \frac{z}{\sin z} \right) \]
by Tsakanikas Nickos
Thu Jul 14, 2016 6:56 pm
Forum: Analysis
Topic: About a $2\pi$ periodical function
Replies: 1
Views: 3082

About a $2\pi$ periodical function

Let \( \displaystyle f \) be a \( \displaystyle C^1 , 2\pi \)-periodical function. If \[ \displaystyle \int_{0}^{2\pi}f(x)\mathrm{d}x = 0 \]show that \[ \displaystyle \int_{0}^{2\pi} \left( f^{\prime}(x) \right)^{2} \mathrm{d}x \geq \int_{0}^{2\pi} \left( f(x) \right)^{2} \mathrm{d}x \] and the equa...
by Tsakanikas Nickos
Thu Jul 14, 2016 6:54 pm
Forum: Complex Analysis
Topic: L'Hôpital's rule
Replies: 0
Views: 1896

L'Hôpital's rule

Let \( \displaystyle f \) and \( \displaystyle g \) be analytic on a region \( \displaystyle A \), both having zeros of order \( \displaystyle k \) at \( \displaystyle z_{0} \in A \). Show that \( \displaystyle \frac{f}{g} \) has a removable singularity at \( \displaystyle z_{0} \) and that \( \disp...
by Tsakanikas Nickos
Thu Jul 14, 2016 2:09 pm
Forum: Calculus
Topic: Evaluation of an integral
Replies: 3
Views: 3293

Evaluation of an integral

Evaluate the integral

\[ \displaystyle \int_{-\infty}^{+\infty} \frac{ \cos(\alpha x) }{1+x^2} \mathrm{d}x \]

where \( \alpha \in \mathbb{R} \smallsetminus \{ 0 \} \).