Asymptotic Expansions of Integrals by Norman Bleistein

By Norman Bleistein

Excellent introductory textual content, written through specialists, offers a coherent and systematic view of ideas and strategies. subject matters include integration through elements, Watson's lemma, LaPlace's strategy, desk bound section, and steepest descents. extra topics include the Mellin rework strategy and not more straightforward elements of the strategy of steepest descents. 1975 edition.

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Dim M. We do it in three steps. D* then the result is Kwack's theorem.

Y - {three points}. 1 of Chapter VII (cf. 10 of Chapter IV, applied in the one-dimensional case. 1. Let X be a relatively compact complex subspace of a complex space Y. Then X is hyperbolically imbedded in Y if and only if, for every length function H on Y there is a constant C > 0 such that for all holomorphic maps f: D -+ X we have f*H ~ CH D • Proof. This is obvious, because over a compact subspace, all length functions are equivalent (each is less than a constant multiple of another). 2. Let X be a complex subspace, relatively compact in Y.

It will suffice to prove that there exists c > 0 such that f(D~) c: U, thus showing that the coordinate functions of f are bounded near 0, so 0 is a removable singularity. Assume there is no such c. After renumbering the sequence, we can assume that f(S(r k») c: U for all k. Let ak , bk be positive numbers with such that the annulus Ak defined by a k < Iz I < b k is the largest annulus whose image under f is contained in U. We let and be the two circles bounding the open annulus. Then but these images of the two circles CX k and 13k are not contained in U.

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