Mathematical Tools for Physics - Department of Physics - University

5—Fourier Series 115

In quantum mechanics, Fourier series and its generalizations will manifest themselves in displaying
the discrete energy levels of bound atomic and nuclear systems.
Music synthesizers areallabout Fourier series and its generalizations.

5.7 Gibbs Phenomenon
There’s a picture of the Gibbs phenomenon with Eq. (5.10). When a function has a discontinuity, its
Fourier series representation will not handle it in a uniform way, and the series overshoots its goal at the
discontinuity. The detailed calculation of this result is quite pretty, and it’s an excuse to pull together
several of the methods from the chapters on series and on complex algebra.

4

π

∑∞

k=0

1

2 k+ 1

sin

(2k+ 1)πx

L

= 1, (0< x < L)

highest harmonic: 5 highest harmonic: 19 highest harmonic: 99
The analysis sounds straight-forward. Find the position of the first maximum. Evaluate the series
there. It really is almost that clear. First however, you have to start with the afinitesum and find the

first maximum of that. Stop the sum atk=N.

4

π

∑N

k=0

1

2 k+ 1

sin

(2k+ 1)πx

L

=fN(x) (5.40)

For a maximum, set the derivative to zero.

fN′(x) =

4

L

∑N

0

cos

(2k+ 1)πx

L

Write this as the real part of a complex exponential and use Eq. (2.3).

∑N

0

ei(2k+1)πx/L=

∑N

0

z^2 k+1=z

∑N

0

z^2 k=z

1 −z^2 N+2

1 −z^2

Factor these complex exponentials in order to put this into a nicer form.

=eiπx/L

e−iπx(N+1)/L−eiπx(N+1)/L

e−iπx/L−eiπx/L

eiπx(N+1)/L

eiπx/L

=

sin(N+ 1)πx/L

sinπx/L

eiπx(N+1)/L

The real part of this changes the last exponential into a cosine. Now you have the product of the sine

and cosine of(N+ 1)πx/L, and that lets you use the trigonometric double angle formula.

fN′ (x) =

4

L

sin 2(N+ 1)πx/L

2 sinπx/L

(5.41)

This is zero at the maximum. The first maximum afterx= 0is at2(N+ 1)πx/L=π, orx=

L/2(N+ 1).