40 1 Mathematical Physics
1.18 The given function is of the square form. Asf(x) is defined in the interval
(−π,π), the Fourier expansion is given by
f(x)=1
2
a 0 +∑∞
n= 1 (ancosnx+bnsinnx)(1)wherean=(1/π)∫π−πf(x) cosnxdx (2)a 0 =(1/π)∫π−πf(x)dx (3)bn=(
1
π)∫π−πf(x)sinnxdx (4)By (3)a 0 =(1/π)(∫ 0
−π0dx+∫π0πdx)
=π (5)By (2)an=(1/π)∫π0cosnxdx= 0 ,n≥1(6)By (4)bn=(1/π)∫π0πsinnx dx=(
1
n)
(1−cosnπ)(7)Using (5), (6) and (7) in (1)f(x)=π
2+ 2
(
sin(x)+(
1
3
)
sin 3x+(
1
5
)
sin 5x+···)
The graph off(x) is shown in Fig. 1.8. It consists of thex-axis from−πto 0
and of the line AB from 0toπ. A simple discontinuity occurs atx =0at
which point the series reduces toπ/2.
Now,
π/ 2 = 1 /2[f(0−)+f(0+)]Fig. 1.8Fourier expansion of
a square wave