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singularities - every neighborhood of z Non-Isolated...

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ACM95a/100a Lecture Notes Niles A. Pierce Caltech 2008 Isolated Singularities The point z 0 is an isolated singularity of f ( z ) if f ( z ) is analytic in a deleted neighborhood 0 < | z - z 0 | < but not at z 0 . Removable Singularities: If f ( z ) has an isolated singularity at z 0 there is an equivalence between: f ( z ) has a removable singularity at z 0 • | f ( z ) | is bounded near z 0 f ( z ) has a (finite) limit as z z 0 f ( z ) can be redefined at z 0 so that f ( z ) is analytic at z 0 Poles: If f ( z ) has an isolated singularity at z 0 there is an equivalence between: f ( z ) has a pole at z 0 • | f ( z ) | → ∞ as z z 0 f ( z ) = g ( z ) / ( z - z 0 ) m for some integer m > 0 and some g ( z ) analytic at z 0 with g ( z 0 ) 6 = 0 Isolated Essential Singularities: If f ( z ) has an isolated singularity at z 0 there is an equivalence between: f ( z ) has an isolated essential singularity at z 0 • | f ( z ) | is neither bounded near z 0 nor goes to infinity as z z 0 f ( z ) assumes every complex number with possibly one exception infinitely many times in
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Unformatted text preview: every neighborhood of z Non-Isolated Singularities Singularities for which f ( z ) is not analytic in a deleted neighborhood of z . Branch Points: The point z is a branch point of f ( z ) if there is a discontinuity in f ( z ) after a small circuit around z . Non-isolated Essential Singularities: The point z is a non-isolated essential singularity of f ( z ) if isolated singularities of f ( z ) cluster around z so that there is no deleted neighborhood of z in which f ( z ) is analytic. Note: Examine singularities at the point at infinity by making the substitution f (1 /w ) and exam-ining w → ....
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