ABSTRACT
In most cases, the series that we shall consider will arise as formal expansions of functions, and will be infinite sums of functions. By a function series we mean an infinite series of the form
aj(x) = a0(x) + a1(x) + a2(x) + · · ·
for some sequence of functions {aj(x)}. Letting sn(x) = ∑n
j=0 aj(x), then the function series converges to s(x) where s(x) = limn sn(x) for all x for which the limit is defined. The set of all x such that the limit exists is called the domain of convergence of the series. Typically, the terms of the series will be drawn from a family of functions of a particular
a0 + a1 (x− c) + a2 (x− c)2 + a3 (x− c)3 + · · · are power series about c. An extended family of mathematical relatives of power series are the series expansions into orthogonal polynomials which have the form
a0 + a1 p1(x) + a2 p2(x) + a3 p3(x) + · · · where pj(x) is a polynomial of degree j. The particular orthogonality condition used to define the polynomials will vary from application to application, but can often be written in the form E [pj(X) pk(X)] = δjk where X is a random variable with given distribution, and δjk is the Dirac delta which equals one or zero as j = k or j = k, respectively. Another important type of series is the asymptotic series, which has the form
a0 + a1 x
+ a2 x2
+ a3 x3
+ · · · which is like a power series, but written out in terms of nonpositive powers of x. We shall encounter the full definition of an asymptotic series in Section 2.4.
