Partial Fractions

Question

Answer 1

(e^(-pi) - 1)/4(s + 1) - 1/[2s(s + 1)]

Consider 2nd term in above expression

1/[2s(s + 1)]   =   A/2s + B/(s + 1)------------------------> (1)

1/[2s(s + 1)]   =   [ A(s + 1) + B(2s) ] / [ 2s(s + 1) ] 

1   =   As + A + 2Bs

0s + 1   =   (A + 2B)s + A

Compare coefficients of above equation

A   =   1

A + 2B   =   0

1 + 2B   =  0

2B   =   -1

B  =  - 1/2

Substitute A = 1, B = -1/2 in equation (1)

1/[2s(s + 1)]   =   (1/2s) + [ (-1/2)/(s + 1) ]

1/[2s(s + 1)]   =   (1/2s) - 1/2(s + 1)

Hence

[ (e^(-pi) - 1)/4(s + 1) ] - 1/[2s(s + 1)]   =   [(e^(-pi) - 1)/4(s + 1)] -  [ (1/2s) - 1/2(s + 1) ]

[ (e^(-pi) - 1)/4(s + 1) ] - 1/[2s(s + 1)]   =   [(e^(-pi) - 1)/4(s + 1)] + 1/2(s + 1) - (1/2s)