Editing Complex analysis (section)

== Major results ==
[[Image:Complex-plot.png|right|thumb|262px|[[Domain coloring|Color wheel graph]] of the function
{{math|''f''(''x'') {{=}} {{sfrac|(''x''<sup>2</sup> − 1)(''x'' − 2 − ''i'')<sup>2</sup>|''x''<sup>2</sup> + 2 + 2''i''}}}}.<br />
[[Hue]] represents the [[Argument (complex analysis)|argument]], [[brightness]] the [[Absolute value#Complex numbers|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 [[residue (complex analysis)|residue]]s 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 theorem#Big Picard|Picard's theorem]]. Functions that have only poles but no [[Essential singularity|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 (complex analysis)|Liouville's theorem]]. It can be used to provide a natural and short proof for the [[Fundamental Theorem of Algebra|fundamental theorem of algebra]] which states that the [[field (mathematics)|field]] of complex numbers is [[algebraically closed field|algebraically closed]].

If a function is holomorphic throughout a [[Connected space|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 [[analytic continuation|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 [[Function of several complex variables|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 Hilbert space|complex space]]s is in [[quantum mechanics]] as [[wave function]]s.