# 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 $U\mathrm{\subset}\mathrm{C}$ be an open domain that contains the origin, and
let $f\mathrm{:}U\mathrm{\to}\mathrm{C}\mathrm{,}$
be a function such that
the complex derivative^{}

$${f}^{\prime}(z)=\underset{\zeta \to 0}{lim}\frac{f(z+\zeta )-f(z)}{\zeta}$$ |

exists for all $z\mathrm{\in}U$. Then, there exists a power series^{} representation

$$ |

for a
sufficiently small radius of convergence^{} $R\mathrm{>}\mathrm{0}$.

Note: it is just as easy to show the existence of a power series representation around every basepoint in ${z}_{0}\in U$; one need only consider the holomorphic function $f(z-{z}_{0})$.

*Proof.* Choose an $R>0$ sufficiently small so that the
disk $\parallel z\parallel \le R$ is contained in $U$.
By the Cauchy integral formula we have
that

$$ |

where, as usual, the integration contour is oriented counterclockwise. For every $\zeta $ of modulus $R$, we can expand the integrand as a geometric power series in $z$, namely

$$ |

The circle of radius $R$ is a compact set; hence
$f(\zeta )$ is bounded^{} on it; and hence, the power series above
converges uniformly with respect to $\zeta $. Consequently, the order
of the infinite^{} summation and the integration operations^{} can be
interchanged. Hence,

$$ |

where

$${a}_{k}=\frac{1}{2\pi i}{\oint}_{\parallel \zeta \parallel =R}\frac{f(\zeta )}{{\zeta}^{k+1}},$$ |

as desired. QED

## Analytic implies holomorphic.

###### Theorem 2

Let

$$ |

be a power series, converging in $D\mathrm{=}{D}_{\u03f5}\mathit{}\mathrm{(}\mathrm{0}\mathrm{)}$, the open disk of radius $\u03f5\mathrm{>}\mathrm{0}$ about the origin. Then the complex derivative

$${f}^{\prime}(z)=\underset{\zeta \to 0}{lim}\frac{f(z+\zeta )-f(z)}{\zeta}$$ |

exists for all $z\mathrm{\in}D$, i.e. the function $f\mathrm{:}D\mathrm{\to}\mathrm{C}$ is holomorphic.

Note: this theorem generalizes immediately to shifted power series in $z-{z}_{0},{z}_{0}\in \u2102$.

*Proof.* For every ${z}_{0}\in D$, the function $f(z)$ can be recast
as a power series centered at ${z}_{0}$. Hence, without loss of
generality it suffices to prove the theorem for $z=0$. The power
series

$$\sum _{n=0}^{\mathrm{\infty}}{a}_{n+1}{\zeta}^{n},\zeta \in D$$ |

converges^{}, and equals
$(f(\zeta )-f(0))/\zeta $ for $\zeta \ne 0$. Consequently, the complex
derivative ${f}^{\prime}(0)$ exists; indeed it is equal to ${a}_{1}$. 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 |