ch9 - Chapter Nine Taylor and Laurent Series 9.1. Taylor...

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Chapter Nine Taylor and Laurent Series 9.1. Taylor series. Suppose f is analytic on the open disk | z z 0 | r .Let z be any point in this disk and choose C to be the positively oriented circle of radius ,where | z z 0 | r . Then for s C we have 1 s z 1 s z 0 z z 0 1 s z 0 1 1 z z 0 s z 0 j 0 z z 0 j s z 0 j 1 since | z z 0 s z 0 | 1. The convergence is uniform, so we may integrate C f s s z ds j 0 C f s s z 0 j 1 ds z z 0 j ,or f z 1 2 i C f s s z ds j 0 1 2 i C f s s z 0 j 1 ds z z 0 j . We have thus produced a power series having the given analytic function as a limit: f z j 0 c j z z 0 j , | z z 0 | r , where c j 1 2 i C f s s z 0 j 1 ds . This is the celebrated Taylor Series for f at z z 0 . We know we may differentiate the series to get f z j 1 jc j z z 0 j 1 9.1
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and this one converges uniformly where the series for f does. We can thus differentiate again and again to obtain f n z j n j j 1  j 2 j n 1 c j z z 0 j n . Hence, f n z 0 n ! c n ,o r c n f n z 0 n ! . But we also know that c n 1 2 i C f s s z 0 n 1 ds . This gives us f n z 0 n ! 2 i C f s s z 0 n 1 ds ,fo r n 0,1,2, . This is the famous Generalized Cauchy Integral Formula. Recall that we previously derived this formula for n 0 and 1. What does all this tell us about the radius of convergence of a power series? Suppose we have f z j 0 c j z z 0 j , and the radius of convergence is R . Then we know, of course, that the limit function f is
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ch9 - Chapter Nine Taylor and Laurent Series 9.1. Taylor...

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