PlanetMath: identity theorem of holomorphic functions

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identity theorem of holomorphic functions

(Theorem)

If the functions $f$ and $g$ are holomorphic in a domain $D$ of the complex plane and the equation

$\displaystyle f(z) = g(z)$

(1)

is true in an infinite subset $S$ of $D$ having an accumulation point $z_0$ in $D$ , then (1) is true in the whole $D$ .

Remark. The subset $S$ may be e.g. some neighbourhood of $z_0$ or some arc containing $z_0$ .

"identity theorem of holomorphic functions" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Other names:

rigidity theorem for analytic functions

This object's parent.

Attachments:: proof of identity theorem of holomorphic functions (Proof) by rspuzio
places of holomorphic function (Corollary) by pahio

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Cross-references: arc, neighbourhood, subset, accumulation point, infinite subset, equation, complex plane, domain, holomorphic, functions
There are 6 references to this entry.

This is version 8 of identity theorem of holomorphic functions, born on 2007-03-04, modified 2008-05-01.
Object id is 9016, canonical name is IdentityTheoremOfHolomorphicFunctions.
Accessed 2319 times total.

Classification:

30A99 (Functions of a complex variable :: General properties :: Miscellaneous)

Pending Errata and Addenda

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