existence of power series
In this entry we shall demonstrate the logical equivalence of the holomorphic
and analytic concepts. As is the case with so many basic results in
complex analysis, the proof of these facts hinges on the Cauchy
integral theorem, and the Cauchy integral formula
.
Holomorphic implies analytic.
Theorem 1
Let be an open domain that contains the origin, and
let
be a function such that
the complex derivative
exists for all . Then, there exists a power series representation
for a
sufficiently small radius of convergence .
Note: it is just as easy to show the existence of a power series representation around every basepoint in ; one need only consider the holomorphic function .
Proof. Choose an sufficiently small so that the disk is contained in . By the Cauchy integral formula we have that
where, as usual, the integration contour is oriented counterclockwise. For every of modulus , we can expand the integrand as a geometric power series in , namely
The circle of radius is a compact set; hence
is bounded on it; and hence, the power series above
converges uniformly with respect to . Consequently, the order
of the infinite
summation and the integration operations
can be
interchanged. Hence,
where
as desired. QED
Analytic implies holomorphic.
Theorem 2
Let
be a power series, converging in , the open disk of radius about the origin. Then the complex derivative
exists for all , i.e. the function is holomorphic.
Note: this theorem generalizes immediately to shifted power series in .
Proof. For every , the function can be recast as a power series centered at . Hence, without loss of generality it suffices to prove the theorem for . The power series
converges, and equals
for . Consequently, the complex
derivative exists; indeed it is equal to . QED
| Title | existence of power series |
|---|---|
| Canonical name | ExistenceOfPowerSeries |
| Date of creation | 2013-03-22 12:56:27 |
| Last modified on | 2013-03-22 12:56:27 |
| Owner | rmilson (146) |
| Last modified by | rmilson (146) |
| Numerical id | 5 |
| Author | rmilson (146) |
| Entry type | Result |
| Classification | msc 30B10 |