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Asymptotics and bounds of the roots of equations. Ed. by L. Eh. Rejzin’. (Ёсимптотика и оценки корней уравнений.) (Russian) Zbl 0731.65034
Riga: Zinatne. 344 p. R. 4.20 (1991).

The book consists of an introduction and four chapters.

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: 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).

Chapter two is devoted to the study of quasipolynomials and the behaviour of their roots – asymptotics of roots, asymptotic expansions of roots, etc.

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, ζ-function, si(x), ci(x), hypergeometric functions, Γ-function, orthogonal polynomials, etc.

Chapter four is devoted to the asymptotics of some functional equations – such as nonautonomous equations, intrinsic functions, iterations, etc.

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.

65H05Single nonlinear equations (numerical methods)
65-02Research monographs (numerical analysis)
65D20Computation of special functions, construction of tables
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
33C10Bessel and Airy functions, cylinder functions, 0 F 1
33C15Confluent hypergeometric functions, Whittaker functions, 1 F 1
33C45Orthogonal polynomials and functions of hypergeometric type
33B15Gamma, beta and polygamma functions
11M06ζ(s) and L(s,χ)