notes5 - The keyhole/Hankel contour and(n Paul Garrett...

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( September 21, 2013 ) The keyhole/Hankel contour and ζ ( - n ) Paul Garrett [email protected] http: / /www.math.umn.edu/ e garrett/ [This document is http://www.math.umn.edu/˜garrett/m/mfms/notes 2013-14/02g keyhole and zeta.pdf] The contour-integration trick illustrated here appeared in one of Riemann’s proofs of analytic continuation of ζ ( s ). It almost immediately proves that values of ζ ( s ) at non-positive integers are rational , and shows the connection to the Laurent coefficients of 1 / ( e t - 1) at t = 0. [1.1] An integral representation of Γ( s ) · ζ ( s ) Although the integral representation of ζ ( s ) using a theta function is perhaps better in the long run, there is a more elementary one: [1.1.1] Claim: For Re( s ) > 1, Γ( s ) · ζ ( s ) = Z 0 t s - 1 dt e t - 1 Proof: Expand a geometric series, exchange sum and integral, and change variables: Z 0 t s - 1 dt e t - 1 = Z 0 t s - 1 e - t dt 1 - e - t = Z 0 t s - 1 X n 1 e - nt dt = X n 1 Z 0 t s e - nt dt t = X n 1 1 n s Z 0 t s e - t dt t = Γ( s ) · X n 1 1 n s = Γ( s ) · ζ ( s ) as claimed. /// [1.2] Keyhole/Hankel contour The keyhole or Hankel contour is a path from + inbound along the real line to ε > 0, counterclockwise around a circle of radius ε at 0, back to ε on the real line, and outbound back to + along the real line. The usual elementary application is to evaluation of integrals similar to R 0 t s dt t 2 +1 , with 0 < Re( s ) < 1. In such an example, analytically continuing counterclockwise around 0 has no impact on the denominator, but,
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