## Asymptotics and Special Functions by F. W. J. Olver

By F. W. J. Olver

A vintage reference, meant for graduate scholars mathematicians, physicists, and engineers, this publication can be utilized either because the foundation for tutorial classes and as a reference device

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J -1 The last integral vanishes when the degree of w(x) is less than n. Therefore φ„(χ) satisfies the orthogonal relation of the Jacobi polynomials. 07) is estabUshed. 09) may be verified in a similar manner. 08) can be used as definitions ofPj^^'^^x) and L^'^^x) for values of α and β for which the orthogonal relations are inapplicable owing to divergence. 10) wix){φ„(x)Ϋdx and the signs of the a„,„. 10). In particular, In a similar way h„ = r{n + oi-^l)/n\ (Laguerre), h„ = n'^^2"nl (Hermite). 13) Of the classical polynomials only the Ll^\x) comprise an orthonormal set.

06) Analytic continuation shows that this result also holds when α is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. Ex. 1 In the notation of §§3 and 4, show that E„(z) = ζ"-'Γ(\-η,ζΙ erfz = n-"^y{^,z^\ erfcz = n-"^T{^,z^l Ex. 2 Show that y(a,z)/{z"r(a)} is entire in α and entire in z, and can be expanded in the form Ex. 3 Show that &·{ζ-'Τ{μ,ζ)}Ι3ζ" = Τ ί α + η,Γ). 1 Let (a,b) be a given finite or infinite interval, and w(x) a function of jc in (a,b) with the properties: (i) w(x) is positive and continuous, except possibly at a finite set of points.

When the boundary points are added the domain is said to be closed; but unless specified otherwise a domain is assumed to be open. 30 1 Introduction to Asymptotic Analysis been given by Davis (1953) and Pittnauer (1969). For discussions of the uniqueness problem see Watson (1911) and Davis (1957). Although such results are of great theoretical interest, practical applications are rare. 3 This generalization is due to Schmidt (1937). For further generalizations of the definition of an asymptotic expansion see Erdelyi and Wyman (1963) and Riekstips (1966).