Home > Holomorphic function

1 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 z₀ of U if the limit

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.

2 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 logarithmIn mathematics, the logarithm functions are the inverses of the exponential functions. Logarithms are numbers that are substituted in computation for other numbers, to which they bear such a relation that the operations to be performed on the latter are r 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}. The inverse trigonometric functions likewise have seams and are holomorphic everywhere except the seams.

3 Properties

Because complex differentiation is linear and obeys the product, quotient, and chain rules, 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 proof that holomorphic functions are analyticComplex analysis Theorems In complex analysis, a complex-valued function f of a complex variable is holomorphic at a point a iff it is differentiable at every point within some open disk centered at a and is analytic at a if in some open disk centered at.

If one identifies C with R², then the holomorphic functions coincide with those functions of two real variables which solve the Cauchy-Riemann equationsPartial differential equations Complex analysis In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. Let f u + iv be a fu, a set of two partial differential equationIn mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. In practice the "unknown function" is usuas.

Close to points with non-zero derivative, holomorphic functions are conformalIn mathematics, a mapping w f ''z is angle-preserving or (more usually) conformal at a point z if it preserves oriented angles between curves through z as well as their orientation, i. Conformal maps preserve both angles and the shapes of infinitesimally in the sense that they preserve angles and the shape (but not size) of small figures.