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See also: holomorphic sheaves and vector bundles.
One central tool in complex analysis is the path integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary ( 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 useful (see methods of contour integration). If a function has a pole or singularity at some point, meaning that its values "explode" and it does not have a finite value there, then one can define the function's residue at that pole, and these residues can be used to compute path integrals involving the function; this is the content of the powerful residue theoremComplex analysis Theorems The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem. The remarkable behavior of holomorphic functions near essential singularities is described by the Weierstrass-Casorati theoremComplex analysis Theorems The Weierstrass-Casorati theorem in complex analysis describes the remarkable behavior of holomorphic functions near essential singularities. Start with an open subset U of the complex plane containing the number z and a holomorp. Functions which have only poles but no essential singularities are called meromorphic.
Laurent seriesIn mathematics, a Laurent series named after Pierre Laurent, is an infinite series. It is a generalization of a power series, but allows terms of negative degree. Definition Specifically, a Laurent series is an infinite series of the form : The numbers a are similar to Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point but can be used to study the behavior of functions near singularities.A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theoremComplex analysis Theorems Liouville's theorem in complex analysis states that every bounded (i. there exists a real number M such that f ''z | ≤ M for all z in C entire function (a holomorphic function f ''z defined on the whole complex plane C must be. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil of complex numbers is algebraically closed.
An important property of holomorphic functions is that if a function is holomorphic throughout a simply 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 where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) are no longer true. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, maybe the most important result in the one-dimensional theory, fails dramatically in higher dimensions.