640 CHAPTER 33. BOUNDARY VALUE PROBLEMS, FOURIER SERIES

which is a nonzero function. Of course any nonzero multiple of this is also an eigenfunc-tion. If µ is not zero, then you need

µL = nπ, n = 1,2, · · ·

so

λ =n2π2

L2 , n = 0,1,2, · · ·

The eigenfunctions in this case are

1, cos(nπ

Lx), n = 1,2, · · ·

Case 3: Next consider the case where y(0) = 0 and y′ (L) = 0.Thus you want nonzeroy and λ such that

y′′+µ2y = 0

y(0) = y′ (L) = 0

In this case, you would have

y =C1 sin µx+C2 cos µx

and on inserting the left boundary condition, this requires that C2 = 0. Now consider theright boundary condition. You can’t have µ = 0 in this case, because if you did, you wouldhave y = 0 which is not allowed. Hence you have

y′ (L) =C1µ cos(µL) = 0

since µ ̸= 0, you must have

µL = (2n−1)π for n = 1,2, · · ·

Therefore, in this case the eigenvalues are

λ =(2n−1)2

π2

L2 , n = 1,2, · · ·

and the eigenfunctions are

sin((2n−1)π

Lx), n = 1,2, · · ·

33.3 Fourier SeriesA Fourier series is a series which is intended to somehow approximate a given periodicfunction by an infinite sum of the form

a0 +∞

∑k=1

ak cos(

kπ

Lx)+

∞

∑k=1

bk sin(

kπ

Lx)

First of all, what is a periodic function?