The topic of precise services is frequently offered as a set of disparate effects, hardly equipped in a coherent means. This booklet emphasizes common rules that unify and demarcate the themes of research. The authors' major pursuits are to supply transparent motivation, effective proofs, and unique references for the entire significant effects. The booklet covers ordinary fabric, but additionally even more. It indicates how a lot of the topic should be traced again to 2 equations - the hypergeometric equation and confluent hypergeometric equation - and it information the ways that those equations are canonical and certain. there's prolonged insurance of orthogonal polynomials, together with connections to approximation concept, persisted fractions, and the instant challenge, in addition to an advent to new asymptotic tools. There also are chapters on Meijer G-functions and elliptic capabilities. the ultimate bankruptcy introduces Painlevé transcendents, which were termed the 'special capabilities of the twenty-first century'.

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