Hurwitz Polynomials

Hurwitz Polynomials

Consider computing the integral where is a Hurwitz polynomial (a polynomial with real positive coefficients and roots that are either negative or pairwise conjugate with negative real parts),

and

From the residue theorem, if and are analytic at and , , and , then has a simple pole at , the residue of at is , and where γ is a simple closed curve enclosing the (see [Marsden 1973, sec. 4.1-3] or [Pearson 1990, sec. 5.4.3].

Hence, the integral reduces to a sum of over the simple poles, that is, the roots of . Michael Trott (trott@wolfram.com) points out that one can use to compute directly.

For example:

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