The Weierstrass Preparation Theorem describes the local structure of a holomorphic function of several complex variables near a point where it vanishes, showing that near such a point the function factors as the product of a nonvanishing holomorphic function and a polynomial of a specific degree in one of the variables, called a Weierstrass polynomial. Named for Karl Weierstrass, it is a foundational tool of several complex variables and analytic geometry, reducing local questions about general holomorphic functions to questions about polynomials.
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StatementA function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at P. 1 Classification
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Source Weierstrass preparation theorem, Wikipedia
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1. Weierstrass preparation theorem, Wikipedia
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a function is, up to multiplication by a function not zero at P, a polynomial in one fixed variable z, which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at P.
In Branch: Complex Analysis, Lead section
Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point
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