Editing Complex analysis (section)

== Complex functions ==
[[Image:Exponentials_of_complex_number_within_unit_circle-2.svg|thumb|right|320px|An [[exponentiation|exponential]] function {{math|''A''<sup>''n''</sup>}} of a discrete ([[integer]]) variable {{mvar|n}}, similar to [[geometric progression]]]]<!-- I detest clueless Fr.wikipedia with their plain-ISO-8859-text-only notation: can somebody find an image of the same, but with a good symbolic notation? -->
A complex function is a [[function (mathematics)|function]] from [[complex number]]s to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as a [[Domain of a function|domain]] and the complex numbers as a [[codomain]]. Complex functions are generally assumed to have a domain that contains a nonempty [[open subset]] of the [[complex plane]].

For any complex function, the values <math>z</math> from the domain and their images <math>f(z)</math> in the range may be separated into [[Real number|real]] and [[Imaginary number|imaginary]] parts:

: <math>z=x+iy \quad \text{ and } \quad f(z) = f(x+iy)=u(x,y)+iv(x,y),</math>

where <math>x,y,u(x,y),v(x,y)</math> are all real-valued.

In other words, a complex function <math>f:\mathbb{C}\to\mathbb{C}</math> may be decomposed into

: <math>u:\mathbb{R}^2\to\mathbb{R} \quad</math> and <math>\quad v:\mathbb{R}^2\to\mathbb{R},</math>

i.e., into two real-valued functions (<math>u</math>, <math>v</math>) of two real variables (<math>x</math>, <math>y</math>).

Similarly, any complex-valued function {{mvar|f}} on an arbitrary [[set (mathematics)|set]] {{mvar|X}} (is [[isomorphic]] to, and therefore, in that sense, it) can be considered as an [[ordered pair]] of two [[real-valued function]]s: {{math|(Re ''f'', Im ''f'')}} or, alternatively, as a [[vector-valued function]] from {{mvar|X}} into <math>\mathbb R^2.</math>

Some properties of complex-valued functions (such as [[continuous function|continuity]]) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as [[differentiability]], are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, every [[holomorphic function|differentiable complex function]] is [[analytic function|analytic]] (see next section), and two differentiable functions that are equal in a [[neighborhood (mathematics)|neighborhood]] of a point are equal on the intersection of their domain (if the domains are [[connected space|connected]]). The latter property is the basis of the principle of [[analytic continuation]] which allows extending every real [[analytic function]] in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number of [[arc (geometry)|curve arc]]s removed. Many basic and [[special functions|special]] complex functions are defined in this way, including the [[exponential function#Complex plane|complex exponential function]], [[complex logarithm|complex logarithm functions]], and [[trigonometric functions#In the complex plane|trigonometric functions]].