Hurwitz Polynomials

Consider computing the integral [Graphics:../Images/tricks_gr_40.gif] where [Graphics:../Images/tricks_gr_41.gif] is a Hurwitz polynomial (a polynomial with real positive coefficients and roots that are either negative or pairwise conjugate with negative real parts),

[Graphics:../Images/tricks_gr_42.gif]

and

[Graphics:../Images/tricks_gr_43.gif]

From the residue theorem, if [Graphics:../Images/tricks_gr_44.gif] and [Graphics:../Images/tricks_gr_45.gif] are analytic at [Graphics:../Images/tricks_gr_46.gif] and [Graphics:../Images/tricks_gr_47.gif], [Graphics:../Images/tricks_gr_48.gif], and [Graphics:../Images/tricks_gr_49.gif], then [Graphics:../Images/tricks_gr_50.gif] has a simple pole at [Graphics:../Images/tricks_gr_51.gif], the residue of [Graphics:../Images/tricks_gr_52.gif] at [Graphics:../Images/tricks_gr_53.gif] is [Graphics:../Images/tricks_gr_54.gif][Graphics:../Images/tricks_gr_55.gif], and [Graphics:../Images/tricks_gr_56.gif]where γ is a simple closed curve enclosing the [Graphics:../Images/tricks_gr_57.gif](see [Marsden 1973, sec. 4.1-3] or [Pearson 1990, sec. 5.4.3].

Hence, the integral [Graphics:../Images/tricks_gr_58.gif] reduces to a sum of [Graphics:../Images/tricks_gr_59.gif] over the simple poles, that is, the roots of [Graphics:../Images/tricks_gr_60.gif]. Michael Trott (trott@wolfram.com) points out that one can use [Graphics:../Images/tricks_gr_61.gif] to compute [Graphics:../Images/tricks_gr_62.gif] directly.

[Graphics:../Images/tricks_gr_63.gif]

For example:

[Graphics:../Images/tricks_gr_64.gif]
[Graphics:../Images/tricks_gr_65.gif]

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