Holomorphic function 

Definition

If U is an open subset of C (see metric space for the definition of "open") and f : U -> C is a function, we say that f is complex differentiable at the point z₀ 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 z₀, and for all such sequences the difference quotient has to approach the same number f '(z₀). Intuitively, if f is complex differentiable at z₀ and we approach the point z₀ from the direction r, then the images will approach the point f(z₀) from the direction f '(z₀) 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 z₀ in U, we say that f is holomorphic on U.

Examples

All polynomial functions with complex coefficients are holomorphic on C, and so are the trigonometric functions 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 logarithm function is holomorphic on the set { z : z isn't a non-positive real number}. The square root function can be defined as

√ z = exp(1/2 ln(z))

Properties

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 doesn't contain 0, even in the vicinity of the negative real line.

If one identifies C with R², then the holomorphic functions coincide with those functions of two real variables which solve the Cauchy-Riemann equations, a set of two partial differential equations.

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 is inside a disk completely determined by its values on the disk's boundary.

• Meromorphic function
• Entire function