In this Appendix—which is lifted essentially verbatim from —we list certain useful relations; we consider their derivation sufficiently straightforward not to require any proof. Not all the formulas reported here are used in this monograph, but we thought it useful to report a rather extensive compilation as a convenient tool for future utilizations.
Let ψ (z, t) be a monic polynomial of degree N in z, and let us indicate as z n ≡ z n (t) its N zeros and as c m ≡ c m (t) its N coefficients:
Here (and throughout) subscripted variables denote partial differentiations,
, and so on.
To conveniently streamline the look of these two, and of all the following, formulas, we rewrite these two relations via the self-evident notation
and we write accordingly the following relations:
To obtain some of the formulas written above we used the relations
which are obvious consequences of the formulas
themselves direct consequences of (A.1
Likewise, we introduce the following notation whereby in the formulas written below (which are also straightforward consequences of (A.1)) the expression in the right-hand side identifies the coefficient of z N −m in the polynomial (of degree N or less) appearing in the left-hand side: