Weierstrass factorization theorem: Difference between revisions

Revision as of 14:47, 14 March 2025

Revision as of 14:49, 14 March 2025

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: <math>f(z)=z^m e^{g(z)} \prod_{n=1}^\infty E_{p_n}\!\!\left(\frac{z}{a_n}\right).</math><ref name="conway">{{citation|last=Conway|first=J. B.|title=Functions of One Complex Variable I, 2nd ed.|publisher=Springer|___location=springer.com|year=1995|isbn=0-387-90328-3}}</ref>

* 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>, <math>m=0</math> and <math>e^{g(z)}=c</math> to obtain: <math>\, f(z) = c\,{\displaystyle\prod}_n (z-a_n)</math>.

====Examples of factorization====