Mathematical Methods for Physics and Engineering : A Comprehensive Guide

12.4 DISCONTINUOUS FUNCTIONS

(a) (b)

(c) (d)

− 1

− 1

− 1

− 1

1

1

1

1

−T 2

−T 2

−T 2

−T 2

T

2

T

2

T

2

T

2

δ

Figure 12.3 The convergence of a Fourier series expansion of a square-wave

function, including (a) one term, (b) two terms, (c) three terms and (d) 20

terms. The overshootδis shown in (d).

arbitrarily close to the discontinuity, it never disappears even in the limit of an

infinite number of terms. This behaviour is known asGibbs’ phenomenon.Afull

discussion is not pursued here but suffice it to say that the size of the overshoot

is proportional to the magnitude of the discontinuity.

Find the value to which the Fourier series of the square-wave function discussed in sec-

tion 12.2 converges att=0.

It can be seen that the function is discontinuous att= 0 and, by the above rule, we expect
the series to converge to a value half-way between the upper and lower values, in other
words to converge to zero in this case. Considering the Fourier series of this function,
(12.8), we see that all the terms are zero and hence the Fourier series converges to zero as
expected. The Gibbs phenomenon for the square-wave function is shown in figure 12.3.