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1: 1.10 Functions of a Complex Variable
Let r 1 = 0 , so that the annulus becomes the punctured neighborhood N : 0 < | z - z 0 | < r 2 , and assume that f ( z ) is analytic in N , but not at z 0 . …Lastly, if a n 0 for infinitely many negative n , then z 0 is an isolated essential singularity. … Suppose F ( z ) is multivalued and a is a point such that there exists a branch of F ( z ) in a cut neighborhood of a , but there does not exist a branch of F ( z ) in any punctured neighborhood of a . …
2: 2.7 Differential Equations
In a punctured neighborhood N of a regular singularity z 0
2.7.4 w j ( z ) = ( z - z 0 ) α j s = 0 a s , j ( z - z 0 ) s , z N ,
2.7.6 w 2 ( z ) = ( z - z 0 ) α 2 s = 0 s α 1 - α 2 b s ( z - z 0 ) s + c w 1 ( z ) ln ( z - z 0 ) , z N .