Uniqueness
Suppose a function ƒ(z) holomorphic on the annulus r < |z − c| < R has two Laurent series:
Multiply both sides with, where k is an arbitrary integer, and integrate on a path γ inside the annulus,
The series converges uniformly on, where ε is a positive number small enough for γ to be contained in the constricted closed annulus, so the integration and summation can be interchanged. Substituting the identity
into the summation yields
Hence the Laurent series is unique.
Read more about this topic: Laurent Series
Famous quotes containing the word uniqueness:
“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)
“Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.”
—Daniel J. Boorstin (b. 1914)