holomorphic function


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holomorphic function

[¦häl·ō¦mȯr·fik ′fəŋk·shən]
(mathematics)
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If f = ([f.sub.1], ..., [f.sub.n]) : D [right arrow] [R.sup.n] is a minimal surface, then there exist holomorphic functions [g.sub.j] in D such that [f.sub.j] = [Reg.sub.j], 1 [less than or equal to] j [less than or equal to] n.
may be considered as inhomogeneous holomorphic function of ([epsilon], z) in S(0, [gamma]; E) x [D.sub.[sigma]](0), and for i = 0 simply (1):
[9-12], for the definition of holomorphic functions see e.g.
In the next proposition, we study the continuity of the operator of multiplication by a holomorphic function.
We shall estimate the growth of the holomorphic function
Since J is holomorphic and [[partial derivative].sub.d]J(c) [not equal to] 0, there exists a holomorphic function g guaranteed by the implicit function theorem such that in some open ball around c, J(X, g(X)) = 0.
Then [S.sub.g,h](C) is identified with the space of holomorphic functions on [[unkown character].sub.g] with the automorphy condition of weight h, under the trivialization by [[omega].sup.h]:
in a Banach space X, which is studied by Gefter and Stulova in [2] under the assumption that A is an invertible closed linear operator with a bounded inverse in X; the delay term h is a complex constant, and f is an X-valued holomorphic function of zero exponential type.
where [mathematical expression not reproducible] is a normalized univalent holomorphic function on D, G is a holomorphic function in [C.sup.n-1] with G(0) = 0, DG(0) = 1, [gamma] [greater than or equal to] 0, and the power functions take the branches such that [mathematical expression not reproducible].
Another interesting considered problem concerns zero sets of holomorphic function in [C.sup.n].
(a) Let f be a holomorphic function in an open domain D and not identically zero.
He shows that the measure is the product of the squared modulus of a holomorphic function with the determinant of the imaginary part of the period matrix to the power of 13.