Ch.5:Series Representations for Analytic Functions Ch.5:Series Representations for Analytic Functions L5.2 Tayfor Series L5.2 Taylor Series Introduction Definition of Taylor Series The n-th-degree polynomial that matches f.ff"...f(n) In Sec.3.1,we learned the Taylor form of the polynomial at zo is Pn(z),centered at zo Pn(2)= Suppose we want to find a polynomial pn(z)of degree at most n that approximates an analytic function f(z)in a o+ae-+re-wr++oe-wr neighborhood of a point zo Definition Naturally there are differing criteria as to how well the If f is analytic at zo,then the series polynomial approximates the function We shall construct a polynomial that "looks like"f(z)at the point zo in the sense that its derivatives match those of f at 、0)+fr0(2-20)+20(-0)2+=0(-0 20 is called the Taylor series for f around zo.When zo =0,it is known as the Maclaurin series of f Ch.5:Series Representations for Analytk Functions Ch.5:Scrics Representations for Analytic Functions 5.2 Taylor Sories -5.2 Taylor Series Convergence of Taylor Series Derivatives of Taylor Series Theorem If f is analytic in the disk z-zo<R,then the Taylor series converges to f(z)for all z in this disk.Furthermore,the Theorem convergence of the series is uniform in any closed subdisk If f is analytic at zo.the Taylor series for f'around zo can be |z-20l≤R<R obtained by termwise differentiation of the Taylor series for f around zo and converges in the same disk as the series for f The theorem implies that the Taylor series will converge to f(z)everywhere inside the largest open disk,centered at zo, over which f is analyticCh.5: Series Representations for Analytic Functions 5.2 Taylor Series Introduction In Sec. 3.1, we learned the Taylor form of the polynomial pn(z), centered at z0 Suppose we want to find a polynomial pn(z) of degree at most n that approximates an analytic function f(z) in a neighborhood of a point z0 Naturally there are differing criteria as to how well the polynomial approximates the function We shall construct a polynomial that ”looks like” f(z) at the point z0 in the sense that its derivatives match those of f at z0 Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Definition of Taylor Series The n-th-degree polynomial that matches f, f, f, ..., f(n) at z0 is pn(z) = f(z0)+f(z0)(z−z0)+f(z0) 2! (z−z0)2+...+f(n)(z0) n! (z−z0)n Definition If f is analytic at z0, then the series f(z0)+f(z0)(z−z0)+ f(z0) 2! (z−z0)2+... = ∞ j=0 f(j)(z0) j! (z−z0)j is called the Taylor series for f around z0. When z0 = 0, it is known as the Maclaurin series of f Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Convergence of Taylor Series Theorem If f is analytic in the disk |z − z0| < R, then the Taylor series converges to f(z) for all z in this disk. Furthermore, the convergence of the series is uniform in any closed subdisk |z − z0| ≤ R < R The theorem implies that the Taylor series will converge to f(z) everywhere inside the largest open disk, centered at z0, over which f is analytic Ch.5: Series Representations for Analytic Functions 5.2 Taylor Series Derivatives of Taylor Series Theorem If f is analytic at z0, the Taylor series for f around z0 can be obtained by termwise differentiation of the Taylor series for f around z0 and converges in the same disk as the series for f