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

By Daniel J. Velleman

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An extension to cotangents and cosecants comes from setting F(sin z, cos z) = cos(z − a1 ) · · · cos(z − a j ) for a generic j < n. We do not recover Theorem 1 if j = n, because then r + s = n, but the argument still works in this case. If F(sin z, cos z) is a homogeneous polynomial of degree n in sin z and cos z, then H (z) is again a cotangent series, and although G(z) no longer approaches zero as the imaginary part of z becomes infinite, it still remains bounded. For future reference, it is convenient to state part of the calculation as a lemma.

K=1 Adding these equations we have 2C = i n + (−i)n , which gives C = cos nπ . 4). Hermite published Theorem 1 in [16], and again a year later in his Cours d’Analyse [17]. Edwards probably learned Hermite’s theorem from [17], although [16] was in an English journal. One does not find [16] in Hermite’s Œuvres, but in its place is [18], which is essentially the same as pp. 320–380 of [17], whereas [16] is more or less the same as pp. 320–340. The relevant parts for us are examples 3 and 4 on pp. 172–73 of [16], pp.

Similarly we can find δ > 0 such that d(x, y) < δ implies d ∗ (x, y) < γ (and hence d(x, y) < ). Suppose that x and y are δ-connected in (X, d). Pick x0 , x1 , . . , xn in X with x0 = x, xn = y, and d(xi , xi+1 ) < δ for i < n. This implies d ∗ (xi , xi+1 ) < γ , so x and y are γ -connected in (X, d ∗ ). But d ∗ is an ultrametric, so repeated application of the ultrametric inequality yields d ∗ (x, y) < γ . Thus d(x, y) < . We can thus infer that if d is uniformly equivalent to an ultrametric then ∀ > 0 ∃ δ > 0 (x, y δ-connected ⇒ d(x, y) < ).

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