Plurisubharmonic function: Difference between revisions

Revision as of 15:31, 23 May 2024

Revision as of 15:33, 23 May 2024

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'''Relation to Dirac Delta:''' On 1-dimensional complex Euclidean space <math>\mathbb{C}^1</math> , <math>u(z) = \log|z|</math> is plurisubharmonic. If <math>f</math> is a C^∞-class function with [[compact support]], then [[Cauchy integral formula]] says

::<math>f(0)=\frac{1}{2\pi i}\int_D\frac{\partial f}{\partial\bar{z}}\frac{dzd\bar{z}}{z},</math>

which can be modified to

::<math>\frac{i}{\pi}\partial\overline{\partial}\log|z|=dd^c\log|z|</math>.

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'''More Examples'''

* If <math>f</math> is an analytic function on an open set, then <math>\log|f|</math> is plurisubharmonic on that open set.

* Convex functions are plurisubharmonic.

* If <math>\Omega</math> is a Domain___domain of Holomorphyholomorphy then <math>-\log (dist(z,\Omega^c))</math> is plurisubharmonic.

==History==