- #1
AStaunton
- 105
- 1
a question on orthogonality relating to Fourier analysis and also solutions of PDEs by separation of variables.
I've used the fact that the following expression (I chose sine, also cosine works):
[tex]\int_{0}^{2\pi}\sin mx\sin nxdx[/tex]
equals 0 unless m=n in which case it equals pi in Fourier analysis and also determing the coefficients of solutions for PDEs by the method of separation of varaibles.
The word orthogonal means perpendicular - what I have never understood is in what sense is sin(mx) perpendicular to sin(nx)?
Also, I have used this orthogonality method when dealing with bessel functions also to collapse a summation to one term as in:
[tex]\int_{0}^{2L}xJ_{0}(\sqrt{\lambda_{n}}x)J_{0}(\sqrt{\lambda_{m}}x)dx[/tex]
where in this problem sqrt(lambda) is eigenvalue. The difference that here when m=n it doesn't evaluate to L as would have been if dealing with trig functions. also had multiply by an extra x as you can see in the above expression...
again my question is, in what sense are the bessel functions perpendicular?
why must multiply the expression by an extra x when dealing with bessels?
and out of interest, does the bessel integral evaluate to something simple when m=n, in the same way that the trig functions evaluate to pi or more generally L?
Be grateful for clarity on these points.
Andrew
I've used the fact that the following expression (I chose sine, also cosine works):
[tex]\int_{0}^{2\pi}\sin mx\sin nxdx[/tex]
equals 0 unless m=n in which case it equals pi in Fourier analysis and also determing the coefficients of solutions for PDEs by the method of separation of varaibles.
The word orthogonal means perpendicular - what I have never understood is in what sense is sin(mx) perpendicular to sin(nx)?
Also, I have used this orthogonality method when dealing with bessel functions also to collapse a summation to one term as in:
[tex]\int_{0}^{2L}xJ_{0}(\sqrt{\lambda_{n}}x)J_{0}(\sqrt{\lambda_{m}}x)dx[/tex]
where in this problem sqrt(lambda) is eigenvalue. The difference that here when m=n it doesn't evaluate to L as would have been if dealing with trig functions. also had multiply by an extra x as you can see in the above expression...
again my question is, in what sense are the bessel functions perpendicular?
why must multiply the expression by an extra x when dealing with bessels?
and out of interest, does the bessel integral evaluate to something simple when m=n, in the same way that the trig functions evaluate to pi or more generally L?
Be grateful for clarity on these points.
Andrew