# holomorphic function

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## hol·o·mor·phic function

(hŏl′ō-môr′fĭk, hō′lō-)
n. Mathematics
A function on a region of a complex plane, differentiable at every point in the region. Also called analytic function.
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Take an arbitrary holomorphic map f: [[DELTA].sup.*] [right arrow] [GAMMA]\D, which extends to a map to a toroidal compactification of [GAMMA]\D.
Now, it is known ([4, 5])--or it could be taken here as a definition--that M is Levi nondegenerate at 0 [member of] M when the local holomorphic map
Let X and Y be complex manifolds of dimensions m and n respectively, and consider a holomorphic map f : X [right arrow] Y.
Let n : Z [right arrow] X be a proper holomorphic map between finite dimensional complex spaces.
: [f.sub.n]] : C [approaches] [P.sub.n] be a holomorphic map of finite order [Lambda].
Therefore both the minimal embedding dimension and the number of elementary matrices needed to factorize a null-homotopic holomorphic map f: X [right arrow] [SL.sub.n](C) is of great interest.
For a fixed i, [g.sub.j]([D.sub.i]) [subset] M for large j, therefore the composition [[pi].sub.1] o [g.sub.j] : [D.sub.i] [right arrow] [omega] is well-defined holomorphic map. By distance decreasing property we have [d.sub.i] = [Mathematical Expression Omitted] for large j.
Let X be a compact Hermitian manifold with the Kaahler form !,and let / : C [right arrow] X be a holomorphic map. We define the spherical derivative [absolute value of df](z)[greater than or equal to]0 by
An holomorphic map f : X [right arrow] Y between two real varieties (X, [[sigma].sub.X]) and (Y, [[sigma].sub.Y]) is called real if it commutes with real structures.
Let f : C [right arrow] [P.sup.n](C) be a holomorphic map. If f(C) omits at least three distinct hyperplanes in [P.sup.n](C) which are linearly dependent over C, then must be degenerate (that is f(C) is contained some proper subspace of [P.sup.n](C)).
Given a holomorphic map f: [C.sup.n] [right arrow] [C.sup.n] with dim [f.sup.-1](0) = O.
Let f : D [right arrow] D' be a proper holomorphic map such that the cluster set [cl.sub.f] (M) [subset] M'.

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