Question

Use method of residues to find the partial fraction decomposition of e^-s+s/s^2+s+1

Answer 1

Use method of residues to find the partial fraction decomposition of \[\frac{e^{-s}+s}{s^2+s+1}\]

Answer 2

not sure how residues does it, but you can go head and use complex numbers to do it

Answer 3

My main problem here is determining how to handle the irreducible quadratic with this method. I know if it was just something like s^2+1 we could split it as (s+i)(s-i). But I keep getting a different equation when I redistribute the factors I try

Answer 4

you can solve using the quadratic formula, or recognize that this is what you get when you solve \[z^3=1\] i.e. \[z^3-1=0\]

Answer 5

solve \[z^3-1=0\] gives
\[(z-1)(z^2+z+1)=0\] the three roots of unity are
\[\large \{1,e^{\frac{2\pi}{3}i},e^{\frac{4\pi}{3}i}\}\]

Answer 6

probably easier to use exponential form than rectangular form, but if you use the quadratic formula you get
\[\frac{-1\pm\sqrt3 i}{2}\] as your zeros
same thing

Answer 7

Okay. I think I understand most of that. Thanks!

Answer 8

yw
if the roots of unity business messed you up, then the quadratic formula always worked
i just recognized that \(z^3-1\) factors as \((z-1)(z^2+z+1)\) so knew what the other two roots are quickly
good luck