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
Famous quotes containing the word remarks:
“The general feeling was, and for a long time remained, that one had several children in order to keep just a few. As late as the seventeenth century . . . people could not allow themselves to become too attached to something that was regarded as a probable loss. This is the reason for certain remarks which shock our present-day sensibility, such as Montaigne’s observation, “I have lost two or three children in their infancy, not without regret, but without great sorrow.””
—Philippe Ariés (20th century)
“There are remarks that sow and remarks that reap.”
—Ludwig Wittgenstein (1889–1951)
“Where do whites fit in the New Africa? Nowhere, I’m inclined to say ... and I do believe that it is true that even the gentlest and most westernised Africans would like the emotional idea of the continent entirely without the complication of the presence of the white man for a generation or two. But nowhere, as an answer for us whites, is in the same category as remarks like What’s the use of living? in the face of the threat of atomic radiation. We are living; we are in Africa.”
—Nadine Gordimer (b. 1923)