Local Behavior
An important topic in potential theory is the study of the local behavior of harmonic functions. Perhaps the most fundamental theorem about local behavior is the regularity theorem for Laplace's equation, which states that harmonic functions are analytic. There are results which describe the local structure of level sets of harmonic functions. There is Bôcher's theorem, which characterizes the behavior of isolated singularities of positive harmonic functions. As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities.
Read more about this topic: Potential Theory
Famous quotes containing the words local and/or behavior:
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“He is not a true man of science who does not bring some sympathy to his studies, and expect to learn something by behavior as well as by application. It is childish to rest in the discovery of mere coincidences, or of partial and extraneous laws. The study of geometry is a petty and idle exercise of the mind, if it is applied to no larger system than the starry one.”
—Henry David Thoreau (1817–1862)