Global Analysis in Linear Differential Equations, Softcover reprint of the original 1st ed. 1999
Mathematics and Its Applications Series, Vol. 471

Since the initiative works for global analysis of linear differential equations by G.G. Stokes and B. Riemann in 1857, the Airy function and the Gauss hypergeometric function became the most important and the greatest practical special functions, which have a variety of applications to mathematical science, physics and engineering. The cffcctivity of these functions is essentially due to their "behavior in the large" . For instance, the Airy function plays a basic role in the asymptotic analysis of many functions arising as solutions of differential equations in several problems of applied math­ ematics. In case of the employment of its behavior, one should always pay attention to the Stokes phenomenon. On the other hand, as is well-known, the Gauss hypergeometric function arises in all fields of mathematics, e.g., in number theory, in the theory of groups and in analysis itself. It is not too much to say that all power series are special or extended cases of the hypergeometric series. For the full use of its properties, one needs connection formulas or contiguous relations.

1 Preparations.- 1.1 Convergent and Divergent Series.- 1.2 Asymptotic Expansions.- 1.3 Linear Difference Equations.- 1.4 Hypergeometric Difference Equation.- 1.5 Modified Gamma Function.- 2 Gauss and Airy Equations.- 2.1 Gauss Equation.- 2.2 Rummer’s Connection Formulas.- 2.3 Monodromy Groups.- 2.4 Associated Fundamental Function.- 2.5 Airy Equation.- 3 Linear Differential Equations.- 3.1 Remarks on Holomorphic Functions.- 3.2 Existence Theorems of Differential Equations.- 3.3 Classification of Singularities.- 3.4 Regular Singular Point.- 3.5 Irregular Singular Point.- 4 Reduction Problems.- 4.1 Reduction to Hypergeometric System.- 4.2 Reduction to Birkhoff Canonical System.- 4.3 Algebraic Manipulation.- 5 Monodromy Groups for Hypergeometric Systems.- 5.1 Extended Gauss Formula.- 5.2 Calculation of Monodromy Groups.- 5.3 Monodromy Group in Logarithmic Case.- 6 Connection Problem for Hypergeometric Systems.- 6.1 General Theory.- 6.2 H. Galbrun Theory.- 6.3 Hierarchy of Connection Coefficients.- 6.4 Jordan-Pochhammer Equation.- 6.5 Appell Hypergeometric Functions.- 6.6 Frobenius Theorem.- 7 Stokes Phenomenon.- 7.1 Two Point Connection Problem.- 7.2 Associated Fundamental Function.- 7.3 Extended Bessel Equation.- 7.4 Derivatives of Stokes Multipliers.- 7.5 Multi-point Connection Problem.