Laurent Series and their Padé Approximations pp 173-185 | Cite as

# Determination of poles

Chapter

## Abstract

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 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

$$\hat b_n^{(m)}$$

*m*→∞ or*m*→−∞. This could be shown by expressing$$\hat b_n^{(m)}$$

*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.## Preview

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© Birkhäuser Verlag Basel 1987