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Additional resources for American Mathematical Monthly, volume 117, number 4, April 2010
3. A. Cayley, Theorem in Trigonometry, Messenger of Math. 5 (1876) 164; also in The Collected Mathematical Papers of Arthur Cayley, vol. 10, Cambridge University Press, Cambridge, 1896. 4. J. , Macmillan, London, 1896. , A Treatise on the Integral Calculus, vol. 1,2, Macmillan, London, 1921, 1922; reprinted by 5. Chelsea, New York, 1954. 6. G. Eisenstein, Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind, und der mit ihnen zusammenh¨angenden Doppelreihen (als eine neue Begr¨undungsweise der Theorie der elliptischen Functionen, mit besonderer Ber¨ucksichtingung ihrer Analogie zu den Kreisfunctionen), J.
An are complex numbers, none of which differs from another by an integer multiple of π. Then all the singularities of G(z) are simple poles at points of the form z = ak + mπ, where 1 ≤ k ≤ n and m is any integer. 6) 1≤ j ≤n j =k = An,k with An,k as in Theorem 1. Hence G(z) behaves like An,k /(z − ak − mπ) near any point of the form z = ak + mπ, in the sense that G(z) − An,k /(z − ak − mπ) has a finite limit there. 3), if we add up all these partial fractions we get n H (z) := ∞ k=1 m=−∞ An,k = z − ak − mπ n An,k cot(z − ak ).
Suppose, for example, that p is fixed, p ≥ 1. The theorem then guarantees that the sequence x defined by p xn = n p−1 (a1 + · · · + anp ) (a1 + · · · + an ) p (n = 1, 2, . . ) (53) must satisfy xn = L np (a) L 1n (a) p ≥ 1. (54) But the first term, x1 , is 1, so that the sequence x, if monotonic, must increase with n. We find, in the same way, that the sequence (53) can only decrease when 0 ≤ p ≤ 1, and increase when p ≤ 0. The fundamental theorem, of course, does not force monotonicity, but it does dictate the directions of any monotonicities that might be contemplated.