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.
Proof. Choose an sufficiently small so that the disk is contained in . By the Cauchy integral formula we have that
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,
as desired. QED
Analytic implies holomorphic.
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|
|Date of creation||2013-03-22 12:56:27|
|Last modified on||2013-03-22 12:56:27|
|Last modified by||rmilson (146)|