Properties
An important property of the set of zeros of a holomorphic function of one variable (that is not identically zero) is that the zeros are isolated. In other words, for any zero of a holomorphic function there is a small disc around the zero which contains no other zeros. There are also some theorems in complex analysis which show the connections between the zeros of a holomorphic (or meromorphic) function and other properties of the function. In particular Jensen's formula and Weierstrass factorization theorem are results for complex functions which have no counterpart for functions of a real variable.
Read more about this topic: Zero (complex Analysis)
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)