Odd and Even functions...

• Aug 25th 2007, 01:28 AM
dudinka
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!
• Aug 25th 2007, 02:31 AM
CaptainBlack
Quote:

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
• Aug 25th 2007, 03:32 AM
dudinka
I'm not sure I got it yet:

1.
Quote:

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!
• Aug 25th 2007, 04:10 AM
CaptainBlack
Quote:

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
• Aug 25th 2007, 04:21 AM
dudinka
10x!
• Aug 25th 2007, 07:59 AM
CaptainBlack
Quote:

Originally Posted by dudinka
10x!

The factorial function does not have any symmetries that I am ware of.

RonL
• Aug 25th 2007, 10:40 AM
dudinka
:-)