Holomorphic Function - Properties

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 not zero.

The derivative can be written as a contour integral using Cauchy's differentiation formula:

for any simple loop positively winding once around, and

for infinitesimal positive loops around .

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

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R2. In other words, if we express a holomorphic function f(z) as u(x, y) + i v(x, y) both u and v are harmonic functions, where v is the harmonic conjugate of u and vice-versa.

In regions where the first derivative is not zero, 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 function holomorphic inside a disk is completely determined by its values on the disk's boundary.

Every holomorphic function is analytic. That is, a holomorphic function f has derivatives of every order at each point a in its domain, and it coincides with its own Taylor series at a in a neighborhood of a. In fact, f coincides with its Taylor series at a in any disk centered at that point and lying within the domain of the function.

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.

From a geometric perspective, a function f is holomorphic at z₀ if and only if its exterior derivative df in a neighborhood U of z₀ is equal to f′(z) dz for some continuous function f′. It follows from

that df′ is also proportional to dz, implying that the derivative f′ is itself holomorphic and thus that f is infinitely differentiable. Similarly, the fact that d(f dz) = f′ dz ∧ dz = 0 implies that any function f that is holomorphic on the simply connected region U is also integrable on U. (For a path γ from z₀ to z lying entirely in U, define

;

in light of the Jordan curve theorem and the generalized Stokes' theorem, F_γ(z) is independent of the particular choice of path γ, and thus F(z) is a well-defined function on U having F(z₀) = F₀ and dF = f dz.)