Formal Definition
Formally, the definition can be stated as follows. Let be a subset of the Euclidean space and let
be an upper semi-continuous function. Then, is called subharmonic if for any closed ball of center and radius contained in and every real-valued continuous function on that is harmonic in and satisfies for all on the boundary of we have for all
Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.
Read more about this topic: Subharmonic Function
Famous quotes containing the words formal and/or definition:
“That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
—Fred Rogers (20th century)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)