Editing Complex number (section)

===Holomorphic functions===
[[File:Sin1z-cplot.svg|thumb|Color wheel graph of the function {{math|sin(1/''z'')}} that is holomorphic except at ''z'' = 0, which is an essential singularity of this function. White parts inside refer to numbers having large absolute values.]]
A function <math>f: \mathbb{C}</math> → <math>\mathbb{C}</math> is called [[Holomorphic function|holomorphic]] or ''complex differentiable'' at a point <math>z_0</math> if the limit
:<math>\lim_{z \to z_0} {f(z) - f(z_0) \over z - z_0 }</math>
exists (in which case it is denoted by <math>f'(z_0)</math>). This mimics the definition for real differentiable functions, except that all quantities are complex numbers. Loosely speaking, the freedom of approaching <math>z_0</math> in different directions imposes a much stronger condition than being (real) differentiable. For example, the function
:<math>f(z) = \overline z</math>
is differentiable as a function <math>\R^2 \to \R^2</math>, but is ''not'' complex differentiable.
A real differentiable function is complex differentiable [[if and only if]] it satisfies the [[Cauchy–Riemann equations]], which are sometimes abbreviated as
:<math>\frac{\partial f}{\partial \overline z} = 0.</math>

Complex analysis shows some features not apparent in real analysis. For example, the [[identity theorem]] asserts that two holomorphic functions {{mvar|f}} and {{mvar|g}} agree if they agree on an arbitrarily small [[open subset]] of <math>\mathbb{C}</math>. [[Meromorphic function]]s, functions that can locally be written as {{math|''f''(''z'')/(''z'' − ''z''<sub>0</sub>)<sup>''n''</sup>}} with a holomorphic function {{mvar|f}}, still share some of the features of holomorphic functions. Other functions have [[essential singularity|essential singularities]], such as {{math|sin(1/''z'')}} at {{math|1=''z'' = 0}}.