Analytic functions
is analytic when for every there is with for ; equivalently .
Definition 3.25 (Analytic Functions)
We say if for every , there exists such that , we have
which is equivalent to saying that
Example
for is analytic on : expanding about , the remainder estimate from the Cauchy form shows for every , that is agrees with its Taylor series throughout .
Vanishing remainders versus convergence
If as , the Taylor series of at may still converge; what fails is that it represents near . Analyticity is exactly the statement that the series converges to , not merely to something.
Related
Stated in
- Definition 3.25 (Analytic Functions)§3.3 Higher Derivatives and Taylor’s Theorem