| n, similar to geometric progression
A complex function is a function from complex numbers to complex numbers. In other words, it is a function that has a subset of the complex numbers as a 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 z=x+iy and f(z)=f(x+iy)=u(x,y)+iv(x,y),
where u, y):
- z0 is defined to be
- Ω can be approximated arbitrarily well by polynomials in some neighborhood of every point in ℂ→ℂ, are holomorphic over the entire complex plane, making them entire functions, while rational functions z↦ℜ(z), z\\mapsto |z|</math>,"}},"i":0}}]}'>f:ℂ→ℂ, defined by f(z) = f(x + iy) = u(x, y) + iv(x, y)</math>,"}},"i":0}}]}'>x,y,u(x,y),v(x,y)∈ℝ, is holomorphic on a region \\Omega</math>,"}},"i":0}}]}'>∂f∂z¯(z0)=0, where ∂∂z¯:=12(∂∂x+i∂∂y).
In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations ,"}},"i":0}}]}'>ℂ∖{z0}, or z0∈ℂ. In other words, if two distinct complex numbers f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. They are a relaxation of the concept of isometries that preserve both angles and lengths.
More formally, let ℝn. A function u0, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size or curvature.
The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors define conformality to include orientation-reversing mappings whose Jacobians can be written as any scalar times any orthogonal matrix.
For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertible complex analytic functions. In three and higher dimensions, Liouville's theorem sharply limits the conformal mappings to a few types.
The notion of conformality generalizes in a natural way to maps between Riemannian or semi-Riemannian manifolds.
 Color wheel graph of the function
2</sup> − 1)(''x'' − 2 − ''i'')<sup>2</sup>|''x''<sup>2</sup> + 2 + 2''i''}}"}},"i":0}}]}'/>f(x) = (x − 1)(x − 2 − i)/x + 2 + 2i.
Hue represents the argument, brightness the magnitude.
One of the central tools in complex analysis is the line integral. The line integral around a closed path of a function that is holomorphic everywhere inside the area bounded by the closed path is always zero, as is stated by the Cauchy integral theorem. The values of such a holomorphic function inside a disk can be computed by a path integral on the disk's boundary (as shown in Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is applicable (see methods of contour integration). A "pole" (or isolated singularity) of a function is a point where the function's value becomes unbounded, or "blows up". If a function has such a pole, then one can compute the function's residue there, which can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by Picard's theorem. Functions that have only poles but no essential singularities are called meromorphic. Laurent series are the complex-valued equivalent to Taylor series, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials.
A bounded function that is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed.
If a function is holomorphic throughout a connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions, such as the Riemann zeta function, which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.
All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension in which the analytic properties such as power series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) do not carry over. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.
A major application of certain complex spaces is in quantum mechanics as wave functions.
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