Toronto Math Forum
APM3462012 => APM346 Math => Misc Math => Topic started by: Chiara Moraglia on November 11, 2012, 03:28:11 PM

Hi, I was just wondering what the symbol O stands for in the notes of lecture 20, for example in formula (10). Thanks!

Hi, I was just wondering what the symbol O stands for in the notes of lecture 20, for example in formula (10). Thanks!
Too bad that in Calculus I you were not given the very standard math notations:
$f=O(g)$ means that $\frac{f}{g}$ is bounded;
$f=o(g)$ means that $\frac{f}{g}$ tends to $0$;
$f\sim g$ means that $\frac{f}{g}$ tends to $1$
and a bit less traditional
$f \asymp g$ means that $c^{1}\le \frac{f}{g}\le c$ (or equivalently $f=O(g)$ and $g=O(f)$).

In the section \textbf{Laplace equation in halfplane} it says
"The problem $u_{yy}+u_{xx}=0$, $y>0$, $u(x,0)=g(x)$ [...] is not uniquely solvable". As an example the function $u(x,y)=y$ is given, which satisfies the Laplace equation. But it does obviously not satisfy $u(x,0)=g(x)$, so I don't see how we can conclude immediately that only bounded solutions are unique...
Can anybode help me with this?
Thanks!

In the section \textbf{Laplace equation in halfplane} it says
"The problem $u_{yy}+u_{xx}=0$, $y>0$, $u(x,0)=g(x)$ [...] is not uniquely solvable". As an example the function $u(x,y)=y$ is given, which satisfies the Laplace equation. But it does obviously not satisfy $u(x,0)=g(x)$, so I don't see how we can conclude immediately that only bounded solutions are unique...
Can anybode help me with this?
Thanks!
It satisfies $u(x,0)=0$; so does trivial solution $u=0$.