1 z2 z2 z2 1z 1z ez z in n is analytic everywhere n z

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Unformatted text preview: −πi+πi = eπi ez −πi = − n=0 The power series is valid everywhere because 12. 1 z2 = z2 = z2 1−z 1−z ez (z − πi)n . n! is analytic everywhere. ∞ n ∞ z= n=0 z n+2 . n=0 The power series is valid on a disc of radius 1 (= distance between 0 and 1, which is the z2 nearest point where is not analytic) 1−z 14. Let f (z ) = Log (1 − z ) . f (0) = Log (1 − 0) = 0 1 = −1 1−0 1 f (0) = − = −1. (1 − 0)2 f (0) = − Log (1 − z ) = f (0) + f (0) 2 f (0) 1 z+ z + . . . = −z − z 2 + . . . ⇒ 1! 2! 2 1 1 −z − z 2 + . . . −z − z 2 + . . . 2 2 14 = z2 + z3 + z + . . . . 4 (Log (1 − z ))2 = 1 The...
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