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| Mathematical analysis →Complex analysis |
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Complex analysis, traditionally known as thetheory of functions of a complex variable, is the branch ofmathematical analysis that investigatesfunctions ofcomplex numbers. It is helpful in many branches of mathematics, includingalgebraic geometry,number theory,analytic combinatorics, andapplied mathematics, as well as inphysics, including the branches ofhydrodynamics,thermodynamics,quantum mechanics, andtwistor theory. By extension, use of complex analysis also has applications in engineering fields such asnuclear,aerospace,mechanical andelectrical engineering.[1]
As adifferentiable function of a complex variable is equal to thesum function given by itsTaylor series (that is, it isanalytic), complex analysis is particularly concerned withanalytic functions of a complex variable, that is,holomorphic functions. The concept can be extended tofunctions of several complex variables.
Complex analysis is an extension ofreal analysis, which deals with the study ofreal numbers andfunctions of a real variable.
Complex analysis is one of the classical branches in mathematics, with roots in the 18th century and just prior. Important mathematicians associated with complex numbers includeEuler,Gauss,Riemann,Cauchy,Weierstrass, and many more in the 20th century. Complex analysis, in particular the theory ofconformal mappings, has many physical applications and is also used throughoutanalytic number theory. In modern times, it has become very popular through a new boost fromcomplex dynamics and the pictures offractals produced by iteratingholomorphic functions. Another important application of complex analysis is instring theory which examines conformal invariants inquantum field theory.
A complex function is afunction fromcomplex numbers to complex numbers. In other words, it is a function that has a (not necessarily proper) subset of the complex numbers as adomain and the complex numbers as acodomain. Complex functions are generally assumed to have a domain that contains a nonemptyopen subset of thecomplex plane.
For any complex function, the values from the domain and their images in the range may be separated intoreal andimaginary parts:
where are all real-valued.
In other words, a complex function may be decomposed into
i.e., into two real-valued functions (,) of two real variables (,).
Similarly, any complex-valued functionf on an arbitrarysetX (isisomorphic to, and therefore, in that sense, it) can be considered as anordered pair of tworeal-valued functions:(Ref, Imf) or, alternatively, as avector-valued function fromX into
Some properties of complex-valued functions (such ascontinuity) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such asdifferentiability, are direct generalizations of the similar concepts for real functions, but may have very different properties. In particular, everydifferentiable complex function isanalytic (see next section), and two differentiable functions that are equal in aneighborhood of a point are equal on the intersection of their domain (if the domains areconnected). The latter property is the basis of the principle ofanalytic continuation which allows extending every realanalytic function in a unique way for getting a complex analytic function whose domain is the whole complex plane with a finite number ofcurve arcs removed. Many basic andspecial complex functions are defined in this way, including thecomplex exponential function,complex logarithm functions, andtrigonometric functions.
Complex functions that aredifferentiable at every point of anopen subset of the complex plane are said to beholomorphic on. In the context of complex analysis, the derivative of at is defined to be[2]
Superficially, this definition is formally analogous to that of the derivative of a real function. However, complex derivatives and differentiable functions behave in significantly different ways compared to their real counterparts. In particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach in the complex plane. Consequently, complex differentiability has much stronger implications than real differentiability. For instance, holomorphic functions areinfinitely differentiable, whereas the existence of thenth derivative need not imply the existence of the (n + 1)th derivative for real functions. Furthermore, all holomorphic functions satisfy the stronger condition ofanalyticity, meaning that the function is, at every point in its domain, locally given by a convergent power series. In essence, this means that functions holomorphic on can be approximated arbitrarily well by polynomials in some neighborhood of every point in. This stands in sharp contrast to differentiable real functions; there are infinitely differentiable real functions that arenowhere analytic; seeNon-analytic smooth function § A smooth function which is nowhere real analytic.
Most elementary functions, including theexponential function, thetrigonometric functions, and allpolynomial functions, extended appropriately to complex arguments as functions, are holomorphic over the entire complex plane, making thementire functions, while rational functions, wherep andq are polynomials, are holomorphic on domains that exclude points whereq is zero. Such functions that are holomorphic everywhere except a set of isolated points are known asmeromorphic functions. On the other hand, the functions,, and are not holomorphic anywhere on the complex plane, as can be shown by their failure to satisfy the Cauchy–Riemann conditions (see below).
An important property of holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as theCauchy–Riemann conditions. If, defined by, where, is holomorphic on aregion, then for all,
In terms of the real and imaginary parts of the function,u andv, this is equivalent to the pair of equations and, where the subscripts indicate partial differentiation. However, the Cauchy–Riemann conditions do not characterize holomorphic functions, without additional continuity conditions (seeLooman–Menchoff theorem).
Holomorphic functions exhibit some remarkable features. For instance,Picard's theorem asserts that the range of an entire function can take only three possible forms:,, or for some. In other words, if two distinct complex numbers and are not in the range of an entire function, then is a constant function. Moreover, a holomorphic function on a connected open set is determined by its restriction to any nonempty open subset.
Inmathematics, aconformal map is afunction that locally preservesangles, but not necessarily lengths.
More formally, let and be open subsets of. A function is called conformal (or angle-preserving) at a point if it preserves angles between directedcurves through, as well as preserving orientation. Conformal maps preserve both angles and the shapes of infinitesimally small figures, but not necessarily their size orcurvature.
The conformal property may be described in terms of theJacobian derivative matrix of acoordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times arotation 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.[3]
For mappings in two dimensions, the (orientation-preserving) conformal mappings are precisely the locally invertiblecomplex 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 betweenRiemannian orsemi-Riemannian manifolds.
One of the central tools in complex analysis is theline 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 theCauchy 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 inCauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory ofresidues among others is applicable (seemethods of contour integration). A "pole" (orisolated 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 powerfulresidue theorem. The remarkable behavior of holomorphic functions near essential singularities is described byPicard's theorem. Functions that have only poles but noessential singularities are calledmeromorphic.Laurent series are the complex-valued equivalent toTaylor series, but can be used to study the behavior of functions near singularities through infinite sums of more well understood functions, such as polynomials.
Abounded function that is holomorphic in the entire complex plane must be constant; this isLiouville's theorem. It can be used to provide a natural and short proof for thefundamental theorem of algebra which states that thefield of complex numbers isalgebraically closed.
If a function is holomorphic throughout aconnected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to beanalytically continued from its values on the smaller domain. This allows the extension of the definition of functions, such as theRiemann 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 thenatural 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 aRiemann surface.
All this refers to complex analysis in one variable. There is also a very rich theory ofcomplex analysis in more than one complex dimension in which the analytic properties such aspower series expansion carry over whereas most of the geometric properties of holomorphic functions in one complex dimension (such asconformality) do not carry over. TheRiemann 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 certaincomplex spaces is inquantum mechanics aswave functions.
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