my teacher wrote that from symmetry considerations we could say that:
F(x)=Sin((n/L)*Pi*x) is an even function for odd n's around X=L/2, and odd for even n's.
could somebody please explain why is that so?
I tried to test it:
x=L/2 => f=Sin(2*Pi*n) and this function is zero at X=L/2 for every n I use. odd or even...!
I'm not sure I got it yet:
1.sin((2n)/L pi (L/2)) = sin (n pi ) -?sin((2n)/L pi (L/2)) = sin( (n pi /2 ) which is 0 for all n
2. Is there a way to notice these qualities quickly?
I mean, he says these things in no time...
Am I missing something? or I should check the qualities with this:
F(L/2-x) = -F(L/2+x) and so on...
(it's related to Fourier analysis of strings modes)