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The Taylor series of a holomorphic function at a point

Definition

Let $f$ be holomorphic on an open set $Ω \subseteq C$ , and let $a \in Ω$ . The Taylor series of $f$ at $a$ is $\sum_{n \geq 0} f^{(n)} (a) (z - a)^{n} / n!$ .

Every derivative $f^{(n)} (a)$ exists by All higher complex derivatives exist and satisfy Cauchy's integral formula on an interior circle, and $n!$ is the positive factorial of The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ , so each coefficient

$c_{n} : = \frac{f^{(n)} (a)}{n!}$

is a well-defined complex number. The resulting complex power series is understood according to Complex series, absolute convergence, complex power series, and radius of convergence. This definition names the formal series; its convergence and equality with $f$ are conclusions of the Taylor expansion theorem.

Depends on

Used by

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27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars Ahlfors, Complex Analysis, 3rd ed., Ch. 5 §1.2 (standard reference, not scraped)
E. Stein and R. Shakarchi, Complex Analysis, Ch. 2 §4 (standard reference, not scraped)
Matthias Weber, Complex Analysis, §2.2 (standard reference, not scraped)
B. V. Shabat, Introduction to Complex Analysis, Ch. 2 (standard reference, not scraped)