notes 10 - Weierstrass and Hadamard products Paul Garrett...

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( August 29, 2013 ) Weierstrass and Hadamard products Paul Garrett [email protected] http: / / e garrett/ [This document is 2013-14/02b Hadamard products.pdf] 1. Weierstrass products 2. Poisson-Jensen formula 3. Hadamard products 4. Appendix: harmonic functions Apart from factorization of polynomials, perhaps the oldest product expression is Euler’s sin πz πz = Y n =1 1 - z 2 n 2 Granting this, Euler equated the power series coefficients of z 2 , evaluating ζ (2) for the first time: π 2 6 = X n =1 1 n 2 The Γ-function factors: Z 0 e - t t z dt t = Γ( z ) = 1 z e γz Y n =1 1 + z n e - z/n where the Euler-Mascheroni constant γ is essentially defined by this relation. The integral (Euler’s) converges for Re( z ) > 0, while the product (Weierstrass’) converges for all complex z except non-positive integers. Granting this, the Γ-function is visibly related to sine by 1 Γ( z ) · Γ( - z ) = - z 2 Y n =1 1 - z 2 n 2 = - z π · sin πz because the exponential factors are linear , and can cancel . Linear exponential factors are exploited in Riemann’s explicit formula [Riemann 1859], derived from equality of the Euler product and Hadamard product [Hadamard 1893] for the zeta function ζ ( s ) = n 1 n s for Re( s ) > 1: Y p prime 1 1 - p - s = ζ ( s ) = e a + bs s - 1 · Y ρ 1 - s ρ e ρs · Y n =1 1 + s 2 n e - s/ 2 n where the product expansion of Γ( s 2 ) is visible, corresponding to trivial zeros of ζ ( s ) at negative even integers, and ρ ranges over all other, non-trivial zeros, known to be in the critical strip 0 < Re( s ) < 1. The hard part of the proof (below) of Hadamard’s theorem is adapted from [Ahlfors 1953/1966], with various rearrangements. A somewhat different argument is in [Lang 1993]. We recall some standard (folkloric?) proofs of supporting facts about harmonic functions, starting from scratch. 1
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Paul Garrett: Weierstrass and Hadamard products (August 29, 2013) 1. Weierstrass products Given a sequence of complex numbers z j with no accumulation point in C , we will construct an entire function with zeros exactly the z j . This is essentially elementary. [1.1] Basic construction Taylor-MacLaurin polynomials of - log(1 - z ) will play a role: let p n ( z ) = z + z 2 2 + z 3 3 + . . . + z n n We will show that there is a sequence of integers n j giving an absolutely convergent infinite product vanishing precisely at the z j , with vanishing at z = 0 accommodated by a suitable leading factor z m : z m Y j 1 - z z j e p n j ( z/z j ) = z m Y j 1 - z z j exp z z j + z 2 2 z 2 j + z 3 3 z 3 j + . . . + z n j n j z n j j Absolute convergence of j log(1 + a j ) implies absolute convergence of the infinite product Q j (1 + a j ). Thus, we show that X j log 1 - z z j + p n j z z j < Fix a large radius R , keep | z | < R , and ignore the finitely-many z j with | z j | < 2 R , so in the following we have | z/z j | < 1 2 . Using the power series expansion of log , log(1 - z z j ) - p n ( z z j ) 1 n + 1 · z z j n +1 + 1 n + 2 · z z j n +2 + . . . 1 n + 1 · | z/z j | n +1 1 - | z/z j | 2 · | z/z j | n +1 n + 1 Thus, we want a sequence of positive integers n j such that X | z j |≥ 2 R | z/z j | n j +1 n j + 1 < (with | z | < R ) Of course, the choice of n j
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  • Spring '14
  • PaulGarrett
  • Math, Complex number, Holomorphic function, Entire function, Hadamard, log |f

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