extra_12 - Extra Problems Sheet 12 Stephen Taylor June 2...

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Stephen Taylor June 2, 2005 1. For each of the following functions, classify the behavior at (i.e., analytic, a zero of order n , pole of order m , or an essential singularity) (a) f ( z ) = z - 1 z +1 . We consider f ( z - 1 ) = 1 - z 1 + z which is analytic at z = 0. So f is analytic at . (b) f ( z ) = z 3 + i z We consider f ( z - 1 ) = 1 z 2 + iz = 1 + iz 3 z 2 which has a pole of order two at 0. So we conclude f has a pole of order 2 at . (c) f ( z ) = z z 3 + i . We consider f ( z - 1 ) = z 2 1 + iz 3 which has a zero of order 2 at 0. So we conclude f has a zero of order 2 at 0. (d) f ( z ) = e z . Since f ( z - 1 ) = e 1 /z , we ﬁnd from the Laurant series expan- sion that this function has an essential singularity at 0. So f has an essential singularity at . (e) f ( z ) = sin z z 2 We note the Laurant series of f ( z - 1 ) has an essential singularity at 0, so f has an essential singularity at . (f)

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extra_12 - Extra Problems Sheet 12 Stephen Taylor June 2...

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