By Norman Bleistein

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**Additional resources for Asymptotic Expansions of Integrals**

**Sample text**

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.