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Zbl 0731.65034
Riekstyn'sh, È.Ya.
(Riekstiņš, E.J.; Reiziņš, L.E. (ed.))
Asymptotics and bounds of the roots of equations. ({\cyr Asimptotika i otsenki kornei0 uravnenii0}.) Ed. by L. Eh. Rejzin'. (Russian)
[B] Riga: Zinatne. 344 p. R. 4.20 (1991). ISBN 5-7966-0570-4

The book consists of an introduction and four chapters. \par Chapter one is a review of some methods and theorems which are used in the numerical treatment of equations of the type $F(x)=0$, $F: \Bbb R\to \Bbb R$ and can be realized in a very general way (with the help of analytical expression, integral, differential equation or whatever). Among these methods are both well known ones and those which can be found only in very special articles (as the usage of integral representations -- for example). \par Chapter two is devoted to the study of quasipolynomials and the behaviour of their roots -- asymptotics of roots, asymptotic expansions of roots, etc. \par Chapter three is an application of the results, obtained in the previous chapter to the asymptotics and root estimation of some special functions - such as cylindrical functions, $\zeta$-function, $\text{si}(x)$, $\text{ci}(x)$, hypergeometric functions, $\Gamma$-function, orthogonal polynomials, etc. \par Chapter four is devoted to the asymptotics of some functional equations -- such as nonautonomous equations, intrinsic functions, iterations, etc. \par The book also contains a lot of examples which are very useful for a student and for an experienced mathematician. The list of references consists of 420 positions.
[Yu.V. Kostarchuk (Chernigov)]

MSC 2000:: *65H05 Single nonlinear equations (numerical methods)
65-02 Research monographs (numerical analysis)
65D20 Computation of special functions
41A60 Asymptotic problems in approximation
33C10 Cylinder functions, etc.
33C15 Confluent hypergeometric functions
33C45 Orthogonal polynomials and functions of hypergeometric type
33B15 Gamma-functions, etc.
11M06 Riemannian zeta-function and Dirichlet L-function

Keywords: zeta function; gamma function; root distribution of special functions; equations; quasipolynomials; asymptotics of roots; asymptotic expansions; cylindrical functions; hypergeometric functions; orthogonal polynomials; nonautonomous equations; intrinsic functions; iterations

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