Part of the Operator Theory: Advances and Applications book series (OT, volume 27)
Determination of poles
The purpose of this chapter is to find the poles of a meromorphic function from its Laurent series expansion in some annular region IK. Again this problem is solved for a McLaurin series of a meromorphic function. The problem is simple if some pole is separated in modules from the moduli of the neighbouring poles. In that case, the pole can be found as the limit of some
parameter for m →∞ or m →−∞. This could be shown by expressing
in terms of Toeplitz determinants as in (9.3g) and using the asymptotics of Toeplitz determinants of chapter 12. If the poles do not separate in modulus, a more complicated situation occurs where Rutishauser polynomials are needed. Again, when m goes to ± ∞, these Rutishauser polynomials will converge to some polynomials whose zeros give a number of the poles of the given meromorphic function. This generalizes the results of theorem 13.1. Similar results for zeros will be derived in the next chapter.
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© Birkhäuser Verlag Basel 1987