For example the functions ex is real analytic at any

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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...
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This document was uploaded on 01/12/2014.

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