New PDF release: Science Without Numbers: A Defence of Nominalism

By Hartry H. Field

ISBN-10: 0691072604

ISBN-13: 9780691072609

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Additional info for Science Without Numbers: A Defence of Nominalism

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53) where the second expression follows from the first by the change of variable x = t−1 . Similarly, ψ(z) − ψ ψ(1 − z) − ψ 1 2 = 1 2 − t−z dt = − 1−t t = 0 t 1 1 x− 2 − xz−1 dx, Re z < 1. 53) gives ψ(z)−ψ(1−z) = − ∞ − 21 t 0 − t−z dt = 1−t ∞ 0 1 x− 2 − xz−1 dx, 0 < Re z < 1. 55) an important relation linking the digamma and the trigonometric functions. 57) by the interchange of z by 1 − z. 52) follows by 1 changing z by − z. This completes the proof. 2 We are now in a position to prove the following remarkable result of Gauss (cf.

M + 1, z), (cf. 1)) m ∈ N, (cf. 17)). The Laurent expansion of ζ(s, a) at s = 1 is given by (cf. 3) ζ(s, a) = ∞ 1 (−1)n γn (a) − ψ(a) + (s − 1)n , s−1 n! n=1 s → 1. 2 n Bk (x) y n−k . 5) March 27, 2007 17:14 WSPC/Book Trim Size for 9in x 6in The theory of the Hurwitz-Lerch zeta-functions vista 55 denote the partial sum of the Hurwitz zeta-function, where for negative values of u, the possible value of n for which n + a = 0 is to be excluded. e. e. all statements about the function in u (Lu (x, a) and ζ(−u, a)) are valid for their derivatives as well in the form of (i) below.

And Γ(u+1−r) interchangeably, where the former is suited for easier calculation and the latter for expected differentiation with respect to u. 1 (Integral Representations) For any l ∈ N with l > Re u + 1 and for any x ≥ 0, we have the integral representation l Lu (x, a) = r=1 Γ(u + 1) (−1)r B r (x) (x + a)u−r+1 Γ(u + 2 − r) r! ∞ (−1)l Γ(u + 1) B l (t) (t + a)u−l dt l! Γ(u + 1 − l) x   1 (x + a)u+1 + ζ(−u, a), u = −1, + u+1 log(x + a) − ψ(a), u = −1.

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Science Without Numbers: A Defence of Nominalism by Hartry H. Field

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