There is one for each square root in the function. it does not necessarily have a (p-)pseudoconvex domain. com; All Rights Reserved.
Numerous further characterizations of such manifolds exist, in particular capturing the property of their having “many” holomorphic functions taking values in the complex numbers. Let E be an open set in Rn, and f be a function that maps E into Rm.
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e. ref 16
From Hartogs’s extension theorem the domain of convergence extends from
H
{\displaystyle H_{\varepsilon }}
to
{\displaystyle \Delta ^{2}}
. ref 83
Grauert introduced the concept of K-complete in the proof of Levi’s problem. Naturally also same function of one variable that depends on some complex parameter is a candidate.
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The spherical harmonics occur in physics and engineering as the solution to Laplace’s equation, as well as the eigenfunctions of the z-component angular momentum operator, which are complex-valued functions of real-valued spherical polar angles:
In quantum mechanics, the wavefunction is necessarily complex-valued, but is a function of real spatial coordinates (or momentum components), as well as time t:
where each is related by a Fourier transform. In the form $f(x,y)=3x+4y-5$ the emphasis has shifted: we now
think of $x$ and $y$ as independent variables and $z$ as a variable
dependent on them, but the geometry is unchanged.
Example 14.
If a function is continuous at f(a), then all the univariate functions that are obtained by fixing all the variables xi except one at the value ai, are continuous at f(a).
A Reinhardt domain D is called a complete Reinhardt domain if together with all point
z
0
D
{\displaystyle z^{0}\in D}
it also contains the polydisc
A complete Reinhardt domain is star-like with respect to its centre a.
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In this situation, we shall also say that X is locally pseudoconvex over Y. This means that if f is differentiable at a point a, then f is continuous at x = a, although the converse is not true – continuity in the domain does not imply differentiability in the domain. Assume that
U
,
V
,
U
V
{\displaystyle U,V,U\cap V\neq \varnothing }
and
{\displaystyle W}
is a connected component of
U
V
{\displaystyle U\cap V}
. .