Taylor series

The power series $\sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$ representing an analytic $f$ near $z_0$ .

The Taylor series of a function $f$ analytic at $z_0$ is the power series $\sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n$ , built from $f$ 's derivatives of every order at $z_0$ . Unlike in real analysis, where a function's Taylor series can fail to converge back to the function, an analytic complex function's Taylor series always converges to $f(z)$ itself throughout the largest disk centered at $z_0$ on which $f$ stays analytic -- analyticity alone guarantees the representation works.

Examples

The Taylor series of $f(z)=e^z$ at $z_0=0$ is $\sum_{n=0}^\infty z^n/n!$ , converging to $e^z$ for every $z$ .
The Taylor series of $f(z)=1/(1-z)$ at $z_0=0$ is $\sum_{n=0}^\infty z^n$ , converging back to $f$ exactly on the disk $|z|<1$ where $f$ is analytic.