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DEFINITION If ''U'' is an : exists. The limit here is taken over all Sequence s of ''complex'' numbers approaching ''z''0, and for all such sequences the difference quotient has to approach the same number ''f'' '(''z''0). Intuitively, if ''f'' is complex differentiable at ''z''0 and we approach the point ''z''0 from the direction ''r'', then the images will approach the point ''f''(''z''0) from the direction ''f'' '(''z''0) ''r'', where the last product is the multiplication of complex numbers. This concept of differentiability shares several properties with Real Differentiability : it is Linear and obeys the product, quotient and chain rules. If ''f'' is complex differentiable at ''every'' point ''z''0 in ''U'', we say that ''f'' is ''holomorphic on U''. We say that ''f'' is holomorphic in the point ''z''0 if it is holomorphic on some neighborhood of ''z''0. We say that ''f'' is holomorphic on some non-open set ''A'' if it is holomorphic in an open set containing ''A''. An equivalent definition is the following. A complex function f(x + iy) = u + iv is holomorphic If And Only If it satisfies the Cauchy-Riemann Equations and u and v have continuous first partial derivatives with respect to x and y. EXAMPLES All Polynomial functions in ''z'' with complex Coefficient s are holomorphic on C, and so are Sine , Cosine and the Exponential Function . (The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's Formula ). The principal branch of the Logarithm function is holomorphic on the Set C - {''z'' ∈ '''R''' : z ≤ 0}. The Square Root function can be defined as : and is therefore holomorphic wherever the logarithm ln(''z'') is. The function 1/''z'' is holomorphic on {''z'' : ''z'' ≠ 0}. Typical examples of functions which are not holomorphic are Complex Conjugation and taking the Real Part . PROPERTIES Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is non-zero. Every holomorphic function is infinitely often complex differentiable at every point. It coincides with its own Taylor Series and the Taylor series converges on every open disk that lies completely inside the domain ''U''. The Taylor series may converge on a larger disk; for instance, the Taylor series for the logarithm converges on every disk that does not contain 0, even in the vicinity of the negative real line. See Holomorphic Functions Are Analytic for a proof. If one identifies C with '''R'''2, then the holomorphic functions coincide with those functions of two real variables which solve the Cauchy-Riemann Equations , a set of two partial Differential Equation s. Close to points with non-zero derivative, holomorphic functions are Conformal in the sense that they preserve angles and the shape (but not size) of small figures. Cauchy's Integral Formula states that every holomorphic function inside a disk is completely determined by its values on the disk's boundary. From an algebraic point of view the set of holomorphic functions on an open set is a Commutative Ring and a Complex Vector Space . In fact, it is a Locally Convex Topological Vector Space , with the seminorms being the suprema on Compact Subsets . SEVERAL VARIABLES A complex analytic function of Several Complex Variables is defined to be analytic and holomorphic at a point if it is locally expandable (within a Polydisk , a Cartesian Product of Disk s, centered at that point) as a convergent power series in the variables. This condition is stronger than the Cauchy-Riemann Equations ; in fact it can be stated as follows: A function of several complex variables is holomorphic If And Only If it satisfies the Cauchy-Riemann equations and is locally Square-integrable . EXTENSION TO FUNCTIONAL ANALYSIS The concept of a holomorphic function can be extended to the infinite-dimensional spaces of Functional Analysis . The article on the Fréchet Derivative reviews the concept of a holomorphic function on a Banach Space . TERMINOLOGY Today, many mathematicians prefer the term "holomorphic function" to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow directly from the definitions. The term "analytic" is however also in wide use. The word "holomorphic" derives from the Greek ''holos'' meaning "whole" and ''morphe'' meaning "form" or "appearance". SEE ALSO |
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