Question

State and prove convolution theorem. Use it to find $L^{-1}\left\{\frac{1}{s(s-3)}\right\}$

Answer 1

Let

Dn=

π

2

−In=∫

π/2

0

f(x)sin(2n+1)x dx

where

f(x)=

1

sinx

−

1

x

.

We need the fact that if we define f(0)=0 then f has a continuous derivative on the interval [0,π/2]. Integration by parts yields

Dn=

1

2n+1

∫

π/2

0

f′(x)cos(2n+1)x dx=O(1/n).

Hence In→π/2 and we conclude that

∫

∞

0

sinx

x

dx= limn→∞ In=

π

2

.