Holomorphic

If a holomorphic function has an infinite sequence of zeroes with an accumulation point, then it is identically zero.

A holomorphic function can be expressed as a convergent power series in a neighborhood of any point in its domain.

If two holomorphic functions with the same domain agree at infinitely many points in a bounded set, then they are identical.

Holomorphic functions have several agreeable properties, making them easy to work with.

If you can show directly that this function's complex derivatives exist everywhere in its domain, then you can conclude that it's holomorphic.

1998, Stefan-M. Heinemann, Julia Sets in Cⁿ, Hartje Kriete, Progress in Holomorphic Dynamics, Addison Wesley Longman, page 159, The iteration of holomorphic maps of one variable is one of the most lively fields of current mathematical research. Since its beginning in the twenties of this century, the theory of holomorphic endomorphisms of the Riemann sphere (= rational functions) has become well developed.

1998, Robert Friedman, Algebraic Surfaces and Holomorphic Vector Bundles, Springer, page 303, A holomorphic 1-form on X is closed, and so 0=dη₁=df and η₂.

The theory of holomorphic dynamical systems is a subject of increasing interest in mathematics, both for its challenging problems and for its connections with other branches of pure and applied mathematics. A holomorphic dynamical system is the datum of a complex variety and a holomorphic object (such as a self-map or a vector field) acting on it.