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The log-Gamma function is an important special function of mathematics, and its principal branch is required in many applications. We develop here the mathematics required to evaluate the principal branch to arbitrary precision, including a new bound for the error in Stirling's asymptotic series. We conclude with a discussion of the implementation of the principal branch of the log-Gamma function in the Maple symbolic algebra system, starting with version Maple V, Release 3. ᮊ 1997 Academic Press

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We show that if X is a separable Banach space such that X' fails the weak* convex point-of-continuity property (C'PCP), then there is a subspace Y of X such that both Y* and {X/Y)* fail C'PCP and both Y and X/Y have finite dimensional Schauder decompositions. It is also an open problem whether for every separable space X with non-separable dual we can find a subspace Y such that both Y" and (X/Y) are non-separable (eventually with finite dimensional decompositions). The purpose of the note is to solve this problem when X has the stronger property that X* does not have the weak* convex point of continuity property.All Banach spaces considered here are real, and are infinite dimensional unless otherwise specified.A dual Banach space X* has the Radon-Nikodym property (RNP in short) if every w* -compact subset C of X* has a point at which the relative weak* and norm topologies coincide [13] and [17].

We introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.