Understanding Engineering Mathematics

In general, we have

an=

1

π

∫π

−π

f(t)cosntdtn= 0 , 1 , 2 ,...

bn=

1

π

∫π

−π

f(t)sinntdtn= 1 , 2 ,...

Problem 17.10

Verify the series for the square wave given in Section 17.9.

We have

f(t)=−A −π<t< 0

=A 0 <t<π

Since the function has period 2π, we can take its series to be:

f(t)=

a 0

2

+

∑∞

n= 1

ancosnt+

∑∞

n= 1

bnsinnt

Also, since the function isoddit can’t contain any even terms in the series, so we can
take allan=0, and write the series as

f(t)=

∑∞

n= 1

bnsinnt

We therefore have to find thebn.
Multiplying through by sinmtand integrating over [−π,π]wehave
∫π

−π

f(t)sinmtdt=

∫π

−π

∑∞

n= 1

bnsinntsinmtdt

=

∑∞

n= 1

bn

∫π

−π

sinntsinmtdt

=bm

∫π

−π

sin^2 mtdt

since ∫π

−π

sinntsinmtdt=0ifm=n

Now ∫π

−π

sin^2 mtdt=

1

2

∫π

−π

( 1 −cos 2mt)dt

=π