# 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 $z_{0}$.

Title identity theorem of holomorphic functions IdentityTheoremOfHolomorphicFunctions 2013-03-22 16:47:05 2013-03-22 16:47:05 rspuzio (6075) rspuzio (6075) 11 rspuzio (6075) Theorem msc 30A99 rigidity theorem for analytic functions IdentityTheoremOfPowerSeries IdentityTheorem