Weierstrass factorization theorem: Difference between revisions

Revision as of 14:34, 14 March 2025

Revision as of 14:45, 14 March 2025

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: <math> \sum_{n=1}^\infty \left( r/|a_n|\right)^{1+p_n} < \infty,</math>

then the function

: <math>fE(z) = \prod_{n=1}^\infty E_{p_n}(z/a_n)</math>

is entire with zeros only at points <math>a_n</math>.<ref name="rudin"/> If a number <math>z_0</math> occurs in the sequence <math>\{a_n\}</math> exactly {{math|''m''}} times, then the function {{math|''fE''}} has a zero at <math>z=z_0</math> of multiplicity {{math|''m''}}.

* The sequence <math>\{p_n\}</math> in the statement of the theorem always exists. For example, we could always take <math>p_n=n</math> and have the convergence. Such a sequence is not unique: changing it at finite number of positions, or taking another sequence {{math|''p''′_''n'' ≥ ''p''_''n''}}, will not break the convergence.

* The theorem generalizes to the following: [[sequences]] in [[open subsets]] (and hence [[Region (mathematics)|regions]]) of the [[Riemann sphere]] have associated functions that are [[Holomorphic function|holomorphic]] in those subsets and have zeroes at the points of the sequence.<ref name="rudin"/>

* Also the case given by the fundamental theorem of algebra is incorporated here. If the sequence <math>\{a_n\}</math> is finite then we can take <math>p_n = 0</math> and obtain: <math>\, f(z) = c\,{\displaystyle\prod}_n (z-a_n)</math>.

===The Weierstrass factorization theorem===