Search found 284 matches
- Wed Jul 20, 2016 6:47 pm
- Forum: Differential Geometry
- Topic: On The Existence Of Shortest Paths
- Replies: 4
- Views: 7357
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 ...
- Wed Jul 20, 2016 10:38 am
- Forum: Differential Geometry
- Topic: On The Existence Of Shortest Paths
- Replies: 4
- Views: 7357
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".
Furthermore, there is also a proof using Arzela-Ascoli's Theorem and the "semicontinuity of the lengths of curves".
- Tue Jul 19, 2016 9:37 pm
- Forum: Differential Geometry
- Topic: On The Existence Of Shortest Paths
- Replies: 4
- Views: 7357
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.
- Fri Jul 15, 2016 6:31 pm
- Forum: Calculus
- Topic: Evaluation of an integral
- Replies: 3
- Views: 3360
Re: Evaluation of an integral
Thank you both for your nice solutions!
- Thu Jul 14, 2016 7:02 pm
- Forum: Complex Analysis
- Topic: Non existence of complex functions
- Replies: 3
- Views: 5124
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|...
- Thu Jul 14, 2016 7:00 pm
- Forum: Complex Analysis
- Topic: Non existence of complex functions
- Replies: 3
- Views: 5124
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...
- Thu Jul 14, 2016 6:58 pm
- Forum: Complex Analysis
- Topic: Evaluation of a limit
- Replies: 0
- Views: 2789
Evaluation of a limit
Evaluate the limit \[ \displaystyle \lim_{z \to 0} \frac{\mathrm{d}^2}{\mathrm{d}z^2} \left( \frac{z}{\sin z} \right) \]
- Thu Jul 14, 2016 6:56 pm
- Forum: Analysis
- Topic: About a $2\pi$ periodical function
- Replies: 1
- Views: 3529
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...
- Thu Jul 14, 2016 6:54 pm
- Forum: Complex Analysis
- Topic: L'Hôpital's rule
- Replies: 0
- Views: 2081
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...
- Thu Jul 14, 2016 2:09 pm
- Forum: Calculus
- Topic: Evaluation of an integral
- Replies: 3
- Views: 3360
Evaluation of an integral
Evaluate the integral
\[ \displaystyle \int_{-\infty}^{+\infty} \frac{ \cos(\alpha x) }{1+x^2} \mathrm{d}x \]
where \( \alpha \in \mathbb{R} \smallsetminus \{ 0 \} \).
\[ \displaystyle \int_{-\infty}^{+\infty} \frac{ \cos(\alpha x) }{1+x^2} \mathrm{d}x \]
where \( \alpha \in \mathbb{R} \smallsetminus \{ 0 \} \).