For example the functions ex is real analytic at any

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: er series for ex , and hence must also be inﬁnite. Power series with positive radius of convergence are so important that there is a special term for describing functions which can be represented by such power series. A function f (x) is said to be real analytic at x0 if there is a power series ∞ an (x − x0 )n n=0 about x0 with positive radius of convergence R such that ∞ an (x − x0 )n , f (x) = for n=0 5 |x − x0 | < R. For example, the functions ex is real analytic at any x0 . To see this, we utilize the law of exponents to write ex = ex0 ex−x0 and apply (1.1) with x replaced by x − x0 : ∞ ex = ex0 ∞ 1 an (x − x0 )n , (x − x0 )n = n! n=0 n=0 where an = ex0 . n! This is a power series expansion of ex about x0 with inﬁnite radius of convergence. Similarly, the monomial function f (x) = xn is real analytic at x0 because n n! xn = (x − x0 + x0 )n = xn−i (x − x0 )i i!(n − i)! 0 i=0 by the binomial theorem, a power series about x0 in which all but ﬁnitely many of the coeﬃcients are zero...
View Full Document

This document was uploaded on 01/12/2014.

Ask a homework question - tutors are online