Holomorphic function
From Academic Kids

Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complexdifferentiable at every point. This is a much stronger condition than real differentiability and implies that the function is infinitely often differentiable and can be described by its Taylor series. The term analytic function is often used interchangeably with "holomorphic function", although note that the former term has several other meanings. A function that is holomorphic on the whole complex plane is called an entire function. The phrase "holomorphic at a point a" means not just differentiable at a, but differentiable everywhere within some open disk centered at a in the complex plane.
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Definition
If U is an open subset of C and f : U > C is a function, we say that f is complex differentiable at the point z_{0} of U if the limit
 <math>f'(z_0) = \lim_{z \rightarrow z_0} {f(z)  f(z_0) \over z  z_0 } <math>
exists. The limit here is taken over all sequences 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}.
Examples
All polynomial functions in z with complex coefficients are holomorphic on C, and so are the trigonometric functions of z 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
 <math>\sqrt{z} = e^{\frac{1}{2}\ln z}<math>
and is therefore holomorphic wherever the logarithm ln(z) is. The function 1/z is holomorphic on {z : z ≠ 0}.
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 nonzero.
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 proof that holomorphic functions are analytic.
If one identifies C with R^{2}, then the holomorphic functions coincide with those functions of two real variables which solve the CauchyRiemann equations, a set of two partial differential equations.
Close to points with nonzero 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 is inside a disk completely determined by its values on the disk's boundary.
Holomorphic functions in 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 disks, centered at that point) as a convergent power series in the variables. This condition is stronger than the CauchyRiemann equations; in fact it can be stated as follows:
A function of several complex variables is holomorphic if and only if it satisfies the CauchyRiemann equations and is locally squareintegrable.