Lecture 44: Residues
Dan Sloughter Furman University Mathematics 39 May 21, 2004
44.1
Some terminology
Recall that we say a point z0 is a singular point of a function f if f is not analytic at z0 but is analytic at some point in every neighborhood of z0 .
Lecture 43: Multiplication and Division of Power Series
Dan Sloughter Furman University Mathematics 39 May 20, 2004
43.1
Multiplication of power series
The following generalization of the power rule is known as Leibnizs rule. Theorem 43.1. If f and g are
Lecture 42: Uniqueness of Series Representations
Dan Sloughter Furman University Mathematics 39 May 19, 2004
42.1
Uniqueness of Taylor series
Theorem 42.1. If f (z ) =
an (z z0 )n
n=0
for all z in an open disk D = cfw_z C : |z z0 | < R, then an = for n =
Lecture 35: Taylor Series
Dan Sloughter Furman University Mathematics 39 May 5, 2004
35.1
Taylor series
Denition 35.1. If f is analytic at a point z0 C, we call the power series f (n) (z0 ) (z z0 )n n!
n=0
the Taylor series of f about z0 . When z0 = 0, we