Weierstrass factorization theorem: Difference between revisions

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===The Weierstrass factorization theorem===

Let {{math|''ƒ''}} be an entire function, and let <math>\{a_n\}</math> be the non-zero zeros of {{math|''ƒ''}} repeated according to multiplicity; suppose also that {{math|''ƒ''}} has a zero at {{math|1=''z'' = 0}} of order {{math|''m'' ≥ 0}} (a zero of order {{math|1=''m'' = 0}} at {{math|1=''z'' = 0}} meansis taken to mean {{math|''&fnof;''(0) ≠ 0}}—that is, <math>f</math> does not have a zero at <math>0</math>).

Then there exists an entire function {{math|''g''}} and a sequence of integers <math>\{p_n\}</math> such that