Math Help - Odd and Even functions...

1. Odd and Even functions...

Hi!

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...!

thanks!

2. Originally Posted by dudinka
Hi!

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...!

thanks!
Odd about L/2 means that F(L/2-x) = -F(L/2+x), and even means that F(L/2-x) = F(L/2+x).

Also sin((2n+1)/L pi (L/2)) = sin( (2n+1)pi /2 ) which is +/-1 for all n, and
sin((2n)/L pi (L/2)) = sin( (n pi /2 ) which is 0 for all n.

RonL

3. I'm not sure I got it yet:

1.
sin((2n)/L pi (L/2)) = sin( (n pi /2 ) which is 0 for all n
sin((2n)/L pi (L/2)) = sin (n pi ) -?

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)

thanks!

4. Originally Posted by dudinka
I'm not sure I got it yet:

1.

sin((2n)/L pi (L/2)) = sin (n pi ) -?

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)

thanks!
In this case its the symmetries of the trig functions that you need to know.

RonL

5. 10x!

6. Originally Posted by dudinka
10x!
The factorial function does not have any symmetries that I am ware of.

RonL

7. :-)