Fourier coefficient calculations

[fluidistic]

Homework Statement

I've been following closely a book on PDE's working through lots of steps but here I'm stuck.
Basically I have the relation [itex]f(x)= \sum _{n=1}^\infty F_n \sin \left ( \frac{n\pi x }{L} \right )[/itex]. I want to calculate the Fourier coefficients [itex]F_n[/itex].
I look at the definition in the same book and I see that if [itex]f(x)= \sum _{n=1}^\infty c_n g_n(x)[/itex] then [itex]c_n=\frac{1}{||g_n||^2} \int _a ^b f(x)g_n (x)dx[/itex].
Since I'm solving the 1 dimensional wave equation between [itex]x=0[/itex] and [itex]x=L[/itex], the limits of the integral are in my case [itex]0[/itex] and [itex]L[/itex].
I get that [itex]F_n =\frac{1}{|| \sin \left ( \frac{n\pi x }{L} \right )||^2} \int _0^L f(x) \sin \left ( \frac{n\pi x }{L} \right ) dx[/itex]. I don't really know how to calculate the modulus squared of the denominator. When I look in the book it says that [itex]F_n=\frac{2}{L} \int _0^L f(x) \sin \left ( \frac{n\pi x }{L} \right ) dx[/itex]. I've absolutely no idea how he did this calculation.

Homework Equations

I don't know if there are any other than the one I posted.

The Attempt at a Solution

Only thoughts so far...
Any clarification is welcome.

 

[jbunniii]

What is the definition of the modulus squared of a function?

 

[fluidistic]

What is the definition of the modulus squared of a function?

Ok thank you, I just found out the result... I didn't know to tell the truth, now I know. Problem solved.