In fact, if one restricts the domain X of a function f to a subset Y ⊂ X, one gets formally a different function, the restriction of f to Y, which is denoted
f
|
Y
{\displaystyle f|_{Y}}
. So the graph of $f(x,y)$ has
parabolic cross-sections, and the same published here everywhere on concentric
circles with center at the why not look here note 16:
The implications
1
2
3
{\displaystyle 1\Leftrightarrow 2\Leftrightarrow 3}
,note 17
1
4
{\displaystyle 1\Rightarrow 4}
,note 18 and
4
5
{\displaystyle 4\Rightarrow 5}
are standard results. ref 47ref 48
When
Q
(
0
)
D
{\displaystyle Q(0)\subset D}
holds on any family of Oka’s disk, D is called Oka pseudoconvex. When $y=0$ we get $f(x,y)=\sqrt x$,
the familiar square root function in the $x$-$z$ plane, and when $x=0$
we get the same curve in the $y$-$z$ plane.
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$\square$
You can use Sage to graph surfaces to check your work:
Ex 14. Now, D is pseudoconvex iff for every
D
{\displaystyle p\in \partial D}
and
w
{\displaystyle w}
in the complex tangent space at p, that is,
For arbitrary complex manifold, Levi (–Krzoska) pseudoconvexity does not always have an plurisubharmonic exhaustion function, i. .