roots α, β
1/(z^2 + bz + 1) = A/(z - α) + B/(z - β)
1/(z^2 + bz + 1)^m = sum (m choose k) [A/(z - α)]^k + [B/(z - β)]^(m-k)
residual of z^(m-1)/(z^2 + bz + 1)^m at α
=
coefficient of 1/(z - α) in z^(m-1)/(z^2 + bz + 1)^m
=
sum (m choose k) (coefficient of 1/(z - α) in z^(m-1) ([A/(z - α)]^k + [B/(z - β)]^(m-k)))
={ k has to be 1 }=
(m choose 1) (coefficient of 1/(z - α) in z^(m-1) ([A/(z - α)]^1 + 0))
=
m z^(m-1) A
={ z is set to the pole α }=
m α^(m-1) A
Comments
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