## Francesco Calogero

Print publication date: 2008

Print ISBN-13: 9780199535286

Published to Oxford Scholarship Online: May 2008

DOI: 10.1093/acprof:oso/9780199535286.001.0001

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# (p.213) APPENDIX A SOME USEFUL IDENTITIES

Source:
Isochronous Systems
Publisher:
Oxford University Press

In this Appendix—which is lifted essentially verbatim from [21]—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 nz n (t) its N zeros and as c mc m (t) its N coefficients:

(A.1)
$Display mathematics$
Then clearly
(A.2)
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(A.3)
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Here (and throughout) subscripted variables denote partial differentiations, $ψ z ≡ ∂ ψ / ∂ z , ψ t ≡ ∂ ψ / ∂ t$, 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

(A.4)
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(A.5)
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and we write accordingly the following relations:
(A.6a)
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(A.6b)
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(A.6c)
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(A.6d)
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(p.214)
(A.7a)
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(A.7b)
$Display mathematics$
(A.8a)
$Display mathematics$
(A.8b)
$Display mathematics$
(A.8c)
$Display mathematics$
(A.8d)
$Display mathematics$
(A.8e)
$Display mathematics$
(A.8f)
$Display mathematics$
(A.8g)
$Display mathematics$
(A.8h)
$Display mathematics$
(A.8i)
$Display mathematics$
(A.8j)
$Display mathematics$
(A.8k)
$Display mathematics$
(p.215)
(A.8l)
$Display mathematics$
(A.8m)
$Display mathematics$
(A.8n)
$Display mathematics$
(A.9a)
$Display mathematics$
(A.9b)
$Display mathematics$
(A.9c)
$Display mathematics$
(A.9d)
$Display mathematics$
(A.9e)
$Display mathematics$
(A.10)
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To obtain some of the formulas written above we used the relations
(A.11)
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(p.216) which are obvious consequences of the formulas
(A.12)
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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:

(A.13a)
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(A.13b)
$Display mathematics$
(A.13c)
$Display mathematics$
(A.13d)
$Display mathematics$
(A.13e)
$Display mathematics$
(A.13f)
$Display mathematics$
(A.13g)
$Display mathematics$
(A.13h)
$Display mathematics$
(A.13i)
$Display mathematics$
(A.13j)
$Display mathematics$
(A.13k)
$Display mathematics$
(A.13l)
$Display mathematics$
(A.13m)
$Display mathematics$
(A.13n)
$Display mathematics$
(A.13o)
$Display mathematics$
(A.13p)
$Display mathematics$
(A.13q)
$Display mathematics$
(A.13r)
$Display mathematics$
(A.13s)
$Display mathematics$
(p.217)
(A.13t)
$Display mathematics$
(A.13u)
$Display mathematics$
(A.13v)
$Display mathematics$