- #1

h_ngm_n

- 4

- 0

I am a bit rusty and have hit a snag with decomposition of partial fractions, I am taking an Engineering course dealing with Laplace transforms. The example is:

F(s)= 3 / s(s

^{2}+2s+5)

Now I get that there are complex roots in the denominator and that there are conjugate complex roots (s+1±2i) giving:

F(s)= 3 / s(s+1-2i)(s+1-2i)

So partial fraction decomposition would give:

F(s) = K

_{1}/s + K

_{2}/(s+1+2i) + K

_{3}/(s+1-2i)

Ok, so solving for K

_{1}= 3/5 is easy and I get that, but when it comes to K

_{2}which involves substituting (-1-2i) in for 's' and then expanding, I can't seem to get the answer. What I did was:

K

_{2}=3[STRIKE](s+1+2i)[/STRIKE]/s(s+1-2i)[STRIKE](s+1+2i)[/STRIKE]

K

_{2}=3/s(s+1-2i)

K

_{2}=3/(-1-2i)[(-1-2i)+1-2i)]

It is the algebraic expansion on the denominator that is getting me. When I do it I get:

(-1-2i)[(-1-2i)+1-2i)]

= (-1-2i)(0-4i)

=-1(-4i)-2i(-4i)

=4i+8i

^{2}

=8+4i

=4(2+i)

However my textbook gets -3/20 (2+i) for the final partial fraction... I'm lost any help would be greatly appreciated.