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Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Like differentiability, convexity is a natural and powerful property of functions that(More)
From an experimental-mathematical perspective we analyze " Ising-class " integrals. These are structurally related n-dimensional integrals we call C n , D n , E n , where D n is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyze C n := 4 n! ∞ 0 · · · ∞ 0 1 n j=1 (u j + 1/u j) 2 du 1 u 1 · · · du n u n. We(More)
Euler considered sums of the form ∞ ∑ m=1 1 m s m−1 ∑ n=1 1 n t. Here natural generalisations of these sums namely [d 1 , d 2 ] := [d 1 , d 2 ](s, t) = ∞ ∑ m=1 χ d 1 (m) m s m−1 ∑ n=1 χ d 2 (n) n t , have been investigated, where χ d 1 and χ d 2 are characters, and s and t are positive integers. The cases when d 1 and d 2 were either 1, 2a, 2b or −4 have(More)
'.. . where almost one quarter hour was spent, each beholding the other with admiration before one word was spoken: at last Mr. Briggs began " My Lord, I have undertaken this long journey purposely to see your person, and to know by what wit or ingenuity you first came to think of this most excellent help unto Astronomy , viz. the Logarithms: but my Lord,(More)
In this paper, properties of the functions A d (q), B d (q) and C d (q) are derived. Specializing at d = 1 and 2, we construct two new quadratic iterations to π. These are analogues of previous iterations discovered by the Bor-weins (1987), (2002). Two new transformations of the hypergeometric series 2 F 1 (1/3, 1/6; 1; z) are also derived.