Cauchy's Integral Formula and Power-Series Expansions

Theorem 1 Assume

f

is analytic on an open set

S

C

, and let

a

be any point of

S

. Then all derivatives

f^{(n)} (a)

exist, and

f

can be represented by the convergent power series

f (z) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (a)}{n!} {(z - a)}^{n},

in every open disc $B (a; R)$ whose closure lies in $S$ . Moreover, for every $n \geq 0$ we have

f^{(n)} (a) = \frac{n!}{2 π i} \int_{γ} \frac{f (w)}{{(w - a)}^{n + 1}} dw,

where $γ$ is any positively oriented circular path with center at $a$ and radius $r < R$ .

Cauchy’s Integral Formula and Power-Series Expansions