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This 3 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH241 at University of Pennsylvania taught by N.Rimmer in Fall. Since its upload, it has received 12 views. For similar materials see /class/215390/math241-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.
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Date Created: 09/28/15
Section 192 Taylor Series A power series with a nonzero radius of convergence represents an analytic function within its circle of convergence Now we want to consider the reverse process Given an can we find its Let f has the series representation Hz M777 m within a 777 gt valld for the largest 777 and I w1th center zo and 777 zo be a 7777 that hes 7777 w1th1n D for zo 0 it is called the 77777 for f centered at zo Sectlan 19 2Twlarnris Familiarfunctions and their Maclaurin series 1zz2z3Zz 1Z n0 2 3 4 on n z z z z z e 1z 2 6 24 quot7012 3 5 m 1quot 2n1 sinzz Z Z ZL 6 120 quot0 2nl 2 4 m 1quot 2n cosz1 Z Z Z Z 2 24 quot0 2n Sectmn 19 ZTmiarsuris Expand in a Maclaurin series and find the radius of convergence fz L 1z2 L lim Ln ngtm an Z Hikm The radius of convergence ofa differentiated series is radius of convergence of the original series Sectlan 19 2Twiarnris Expand in a Taylor series centered at zaand find the radius of convergence 1 want fz3 z 1p0wersof Z 2i 2 1 1 th igti1E E TE Ti I I M a u Q n1 L1im ngtm L n 5mm 9 gamma A point z z0 where a complex function fails to be analytic is called a or of the function m R z z21 0 are singularities off A point z z0 is called an of a function if there exists some quotdeletedquot neighborhood or quotpuncturedquot open disk of zo 0 lt z zol lt R where the function is Bothi and 7i are isolated singularities off 3Z1 since z f is analytic on the deleted neighborhood and is the distance from the center zo of the series to the nearest c 0 R z02i
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