Weierstrass factorization theorem: Difference between revisions

Revision as of 08:47, 30 October 2021

Revision as of 13:19, 19 February 2022

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===Hadamard factorization theorem===

If {{math|''ƒ''}} is an entire function of finite [[Entire function|order]] {{math|''ρ''}} and {{math|''m''}} is the order of the zero of {{math|''ƒ''}} at {{math|1=''z''}} ={{math|'' 0''}}, then it admits a factorization

: <math>f(z) = z^m e^{g(z)} \displaystyle\prod_{n=1}^\infty E_{p}\!\!\left(\frac{z}{a_n}\right)</math>

where {{math|''g''(''z'')}} is a polynomial of degree {{math|''q''}}, {{math|''q'' ≤ ''ρ''}} and {{math|1=''p'' = [''ρ'']}} is the integer part of {{math|''ρ''}}.<ref name="conway" />