Laurent Series Obtained by Long Division

Abstract

<p>Let r_1,...,r_nbe the n root-moduli of the polynomial azⁿ+bz^m+c, where n>m>0 are integers and a,b,c are nonzero complex numbers. We give a necessary and sufficient condition in order that the long division of <strong>^.</strong>1 by bz^m+azⁿ+c (where contrary to traditional long division, the divisor is ordered neither in the ascending nor in the descending powers of z) yield the Laurent series of 1/(azⁿ+bz^m+c) valid in the annulus r_k< IzI k+1 </sub>for some root-modulus r_k. Our method gives an effective way of obtaining Laurent series of 1/(azⁿ+bz^m+c) in nontrivial annulus requiring no information about the roots of azⁿ+bz^m+c. Our method can be generalized to yield Laurent series of P(z)/Q(z) in all pertinent nontrivial annuli, where P(z) and Q(z) are any finite (or infinite) polynomials. The generalization consists of (possible premultiplication of the numerator and the denominator of P(z)/Q(z) by a suitable polynomial) choosing as the leading term for long division a suitable split of a suitable term in the (possibly new) denominator.</p>