TABLE OF CONTENTS One of the most powerful mathematical tools in the physicist s armory is expansion of an analytic ( smooth ) function in a series of ascending powers of a small parameter. While infinite series form an important part of any undergraduate mathematical curriculum, a physicist looks at series from a point of view somewhat different than that of most mathematicians.
Suppose that we have analyzed some physical phenomenon and have found a function that describes it. Initially we will assume that it is a function of a single variable which we call f(x) (Section A.5 extends the analysis to functions of multiple variables). Often this function is inconveniently complicated in general, but is simple for a certain special value of its argument. Series expansion provides a simple expression for its behavior close that value. As we shall see, the question of convergence, which dominates the discussion of infinite series in most textbooks of mathematics, is rarely important in physics.
By the right choice of variable, we can take the special value of x to be zero with no loss of generality, and we assume that f(x) is analytic in the region x=0; a function is analytic in a certain domain if all its derivatives exist in that domain. This is not true of all functions; for example x a is not analytic at x=0 unless a is a non-negative integer, and...
