Mathematical Tools for Physics

5—Fourier Series 123

cosnπx/L (n= 0, 1 , 2 ,...), or you can choose a basis

sin(n+^1 / 2 )πx/L (n= 0, 1 , 2 ,...), or you can choose a basis

e^2 πinx/L (n= 0,± 1 ,± 2 ,...), or an infinite number of other possibilities.

Fundamental Theorem
If you want to show that each of these respective choices provides an orthogonal set of functions you can do a
lot of integration or you can do all the cases at once with an important theorem that starts from the fact that all
of these sines and cosines and complex exponentials satisfy the same differential equation,u′′=λu, whereλis
some constant, different in each case.
You have two functionsu 1 andu 2 that satisfy

u′′ 1 =λ 1 u 1 and u′′ 2 =λ 2 u 2

Make no assumption about whether theλ’s are positive or negative or even real. Theu’s can also be complex.
Multiply the first equation byu 2 and the second byu 1 , then take the complex conjugate of the second product.

u* 2 u′′ 1 =λ 1 u* 2 u 1 and u 1 u∗′′ 2 =λ* 2 u 1 u* 2

Subtract the equations.
u 2 u′′ 1 −u 1 u∗′′ 2 = (λ 1 −λ 2 )u* 2 u 1

Integrate fromatob ∫
b

a

dx

(

u* 2 u′′ 1 −u 1 u∗′′ 2

)

= (λ 1 −λ* 2 )

∫b

a

dxu* 2 u 1 (11)

Now do some partial integration. Work on the second term on the left:

∫b

a

dxu 1 u∗′′ 2 =u 1 u∗′ 2

∣

∣

∣

∣

b

a

−u′ 1 u* 2

∣

∣

∣

∣

b

a

+

∫b

a

dxu′′ 1 u* 2

Put this back into the Eq. ( 11 ) and the integral terms on the left cancel.

u′ 1 u* 2 −u 1 u∗′ 2

∣

∣

∣

∣

b

a

= (λ 1 −λ* 2 )

∫b

a

dxu* 2 u 1 (12)