Mapping Diagram Visualizing Complex Derivative for Powers

[b][center]The Complex Derivative and the Differential[/center]Suppose [math]U\subset\mathbb{C}[/math] is a domain, [math]z_0\in U[/math], and [math]f:U\longrightarrow\mathbb{C}[/math].[br][br]Definition: [/b]The [b]derivative[/b] of [math]f[/math] at [math]z_0[/math], denoted [math]f'(z_0)[/math], is defined by[br] [math]f'(z_0)\equiv\lim_{z\to z_0}\frac{f(z)-f(z_0)}{z-z_0}[/math] [b]provided the limit exists.[br][/b]If [math]f'(z)[/math] exists for all [math]z[/math] in some open set containing [math]z_0[/math] , [math]f[/math] is called [b]holomorphic[/b] at [math]z_0[/math].[br][br][b]Definition: [/b]Suppose [math]f[/math] is holomorphic at [math]z_0[/math].[br]The [b]differential[/b] of [math]f[/math] at [math]z_0[/math] for [math] \Delta z\in\mathbb{C}[/math] , denoted [math]df(z_0,\Delta z)[/math], is defined by [math]df(z_0,\Delta z)=f'(z_0)\Delta z[/math] .[br][br][b]Fact: [/b] If [math]f[/math] is holomorphic at [math]z_0[/math] and [math]z_0+\Delta z\in U[/math], then [math]f(z_0+\Delta z)\approx f(z_0)+df(z_0,\Delta z)[/math].