Remarks
- Since power series can be differentiated term-wise, applying the above argument in the reverse direction and the power series expression for
- gives
- This is a Cauchy integral formula for derivatives. Therefore the power series obtained above is the Taylor series of ƒ.
- The argument works if z is any point that is closer to the center a than is any singularity of ƒ. Therefore the radius of convergence of the Taylor series cannot be smaller than the distance from a to the nearest singularity (nor can it be larger, since power series have no singularities in the interiors of their circles of convergence).
- A special case of the identity theorem follows from the preceding remark. If two holomorphic functions agree on a (possibly quite small) open neighborhood U of a, then they coincide on the open disk Bd(a), where d is the distance from a to the nearest singularity.
Read more about this topic: Analyticity Of Holomorphic Functions
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