I want to derive Laurent series of f ( z ) = 1 ( z

Jonathan Kent 2022-05-23 Answered
I want to derive Laurent series of
f ( z ) = 1 ( z 2 + 1 ) ( z 2 9 )
for two sets: 1 < | z | < 3 and 3 < | z |
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Answers (1)

Conor Frederick
Answered 2022-05-24 Author has 7 answers
For Laurent series, it is useful to remember the following two geometric series,
1 1 c z = n = 0 ( c z ) n ,  for  | z | < 1 | c | , 1 1 b z = n = 0 ( b z ) n ,  for  | z | > | b | .
Note that 1 z 2 1 = 1 z i + 1 z + i . So we can find the Laurent series for each of these.
1 z i = 1 z 1 1 i z = 1 z n = 0 ( i z ) n ,  for  | z | > 1 = | i | = n = 0 i n z n + 1 ,  for  | z | .
Then the others can be calculated in a similar way, using the geometric series.

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