1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Easy partial fractions explanation

  1. Aug 8, 2007 #1
    1. The problem statement, all variables and given/known data

    I just want to know how to proceed to get
    1/s - s/(s^2+1)

    using partial fractions on the term

    1/(s(s^2 − 1))

    I know this is probably straight forward but I just don't get it.

    Thanks.
     
  2. jcsd
  3. Aug 8, 2007 #2

    malawi_glenn

    User Avatar
    Science Advisor
    Homework Helper

    [tex] \dfrac{1}{s(s^2-1)} = \dfrac{A}{s} + \dfrac{B}{s-1} + \dfrac{C}{s+1} [/tex]

    Since s^2 - 1 = (s+1)(s-1)

    now get LHS = RHS (left hand side, right hand side)

    [tex] \dfrac{1}{s(s^2-1)} = \dfrac{A(s-1)(s+1)}{s(s-1)(s+1)} + \dfrac{Bs(s-1)}{s(s-1)(s+1)} } + \dfrac{Cs(s+1)}{s(s-1)(s+1)} } = \dfrac{A(s-1)(s+1) + Bs(s-1) + Cs(s+1)}{s(s^2-1)}[/tex]

    Now do all the multiplication on the RHS, and solve the linear equation system with respect to A,B and C.
     
  4. Aug 8, 2007 #3

    HallsofIvy

    User Avatar
    Science Advisor

    You DON'T. (And so it is certainly not "straight forward"!)

    1/s- s/(s^2+ 1) would come from something with denominator s(s^2+1)= s^3+ 1.

    1/(s(s^2-1)= 1/(s(s-1)(s+1)) gives partial fractions of the form
    A/s+ B/(s-1)+ C/(s+1)

    Malawi_glenn showed one way to do that. I would have done it slightly differently.
    Write 1/(s(s-1)(s+1))= A/s+ B/(s-1)+ C/(s+1) and, instead of adding the fractions on the right, get rid of the fractions by multiplying both sides by s(s-1)(s+1). That gives 1= A(s-1)(s+1)+ Bs(s+1)+ Cs(s-1).
    Letting s= 0 in that gives 1= A(-1)(1)= -A so A= -1.
    Letting s= 1 in that gives 1= B(1)(2)= 2B so B= 1/2.
    Letting s= -1 in that gives 1= C(-1)(-2)= 2C so C= 1/2
    1/(s(s^2-1))= -1/s+ (1/2)/(s-1)+ (1/2)/(s+1).

    Now, if you meant 1/s(s^2+1)) orginally, then you know that
    1/s(s^2+1))= A/s+ (Bs+ C)/(s^2+1)

    We can, again, eliminate the fractions by multiplying both sides by s(s^2+ 1) to get 1= A(s^2+1)+ (Bs+ C)s

    Taking s= 0 gives 1= A so A= 1

    Unfortunately, there is no real s that makes s^2+ 1= 0 so just take s= 1 and -1 to get 1= (1)(2)+ (B(1)+C)(1) and 1= (1)(2)+ (B(-1)+ C)(-1) or
    B+C= -1 and B- C= -1. Adding those two equations 2B= -2 so B= -1 and then -1+ C= -1 so C= 0.

    1/(s(s^2+ 1))= 1/s- s/(s^2+ 1)

    Was that what you meant?
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook