Download American Mathematical Monthly, volume 117, number 4, April by Daniel J. Velleman PDF

By Daniel J. Velleman

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28. , Introductio in Analysin Infinitorum, vol. 1, Marcum-Michaelum Bousquet, Lausanne, 1748; also Opera Omnia, ser. I, vol. 8, Teubner, Leipzig, 1922; English translation by J. Blanton, Springer-Verlag, New York, 1988. J. Glaisher, Note on the above, Messenger of Math. 8 (1878) 46–47. , Trigonometrical theorems involving the products of four sines or four cosines, Messenger of Math. 10 (1880) 26–34. J. Hadamard, The Mathematician’s Mind, Princeton University Press, Princeton, NJ, 1996; reprint of The Psychology of Invention in the Mathematical Field, Princeton University Press, Princeton, NJ, 1945.

J. Hadamard, The Mathematician’s Mind, Princeton University Press, Princeton, NJ, 1996; reprint of The Psychology of Invention in the Mathematical Field, Princeton University Press, Princeton, NJ, 1945. G. H. Hardy, A Course of Pure Mathematics, Centenary Edition, Cambridge University Press, Cambridge, 2008. ∞ , The integral 0 sinx x d x, Math. Gaz. 5 (1909) 98–103; also in Collected Papers of G. H. Hardy, vol. 5, Clarendon Press, Oxford, 1972. 2307/3602798 ∞ , Further remarks on the integral 0 sinx x d x, Math.

1≤i< j ≤n . u n vnn−2 vnn−1 This is easy to derive from the ordinary Vandermonde determinant, in which u i = 1 for every i, and it is admirably suited to a trigonometric change of variables: if we set u i = sin ai and vi = cos ai for each i we obtain sinn−1 a1 sinn−1 a2 sinn−1 a3 .. sinn−1 an sinn−2 a1 cos a1 sinn−2 a2 cos a2 sinn−2 a3 cos a3 .. sinn−2 an cos an ··· ··· ··· .. ··· sin a1 cosn−2 a1 sin a2 cosn−2 a2 sin a3 cosn−2 a3 .. cosn−1 a1 cosn−1 a2 cosn−1 a3 .. 4) sin ai − a j , = 1≤i< j ≤n which was also first given by van den Berg [1, p.

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