[b]Taylor's theorem[/b] generalizes to functions[math]f:U\longrightarrow\mathbb{C}[/math] which are [url=https://en.wikipedia.org/wiki/Complex_differentiable]complex differentiable[/url] in an open subset [math]U⊂\mathbb{C}[/math] of the [url=https://en.wikipedia.org/wiki/Complex_plane]complex plane[/url].[br][br]The [b]Taylor polynomial[/b] holds in the form similar to that for real analysis: [br][math]f(z)=P_k(z)+R_k(z)[/math] , [math]P_k(z)=\sum_{j=0}^k \frac {f^{(j)}(c)}{j!}(z-c)^j[/math] , and [math]R_k(z)=f(z)-P_k(z)[/math],[br]which the following figure visualizes and compares using mapping diagrams restricted to circles.