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# identity theorem of holomorphic functions

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 $z_{0}$.

Related:

IdentityTheoremOfPowerSeries, IdentityTheorem

Synonym:

rigidity theorem for analytic functions

Major Section:

Reference

Type of Math Object:

Theorem

Parent:

## Mathematics Subject Classification

30A99*no label found*

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