Mathematical Tools for Physics

5—Fourier Series 134

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.

5.6 Gibbs Phenomenon
There’s a picture of the Gibbs phenomenon with Eq. ( 7 ). 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 afinite sum 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) (32)

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. (2.2).

∑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