Orthogonal Properties for Sine Don't Hold if Pi is involded?

[mmmboh]

Orthogonal Properties for Sine Don't Hold if Pi is involded??

Normally I know <br /> \int_{-L}^L \sin \frac{n x}{L} \sin \frac{\m x}{L} ~ dx = 0\mbox{ if }n\not =m , \ =L \mbox{ if }n=m<br /> but apparently this doesn't work for <br /> \int_{-L}^L \sin \frac{\pi n x}{L} \sin \frac{\pi m x}{L} ~ dx <br />

I am trying to find Fourier coefficients, and my integral is \int_{0}^{L/2} \sin \frac{2\pi x}{L} \sin \frac{\pi n x}{L} ~ dx <br />

For even n that aren't 2, the integral is 0, if n=2, then the integral is L/2, but if n is odd, then the integral doesn't equal 0 (it's actually a fairly complex answer) even thought it looks like it should by orthogonal properties ...why doesn't this work?

The integrals I am doing take very long to do, is there a property that would allow me to shorten them?

 

[LCKurtz]

It does work. If n \ne m are positive integers then

\int_0^L\ \sin(\frac{n\pi x}{L})\sin(\frac{m\pi x}{L})\, dx = 0

Sturm-Liouville theory would guarantee this without working it out.

 

[mmmboh]

Sorry the limits are actually 0 to L/2, in this case it doesn't work, is there another property that would help evaluate the integral quicker or do I have to go through the whole thing every time?

 

[LCKurtz]

The set of functions

\left \{ \sin\frac{2n\pi x}{L}\right\}

are orthogonal on [0, L/2].

 

[mmmboh]

Is there anything like that for <br /> \left \{ \sin\frac{(2n)\pi x}{L}\right\ \left \sin\frac{(2n+1)\pi x}{L}\right\}<br />?

 

[LCKurtz]

Is there anything like that for <br /> \left \{ \sin\frac{(2n)\pi x}{L}\right\ \left \sin\frac{(2n+1)\pi x}{L}\right\}<br />?

I doubt you will get zero, but why don't you just integrate it and see?