lecture15 - 8 Laurent series Theorem(Laurents...

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8. Laurent series Theorem (Laurent’s Theorem). [S&T11.1] If f is differentiable in the annulus { z C : R 1 ≤ | z z 0 | ≤ R 2 } , where 0 R 1 < R 2 ≤ ∞ , then f ( z ) = n =0 a n ( z z 0 ) n + n =1 b n ( z z 0 ) - n , where n =0 a n ( z z 0 ) n converges for | z z 0 | < R 2 and n =1 b n ( z z 0 ) - n converges for | z z 0 | > R 1 . In particular, both series converge in the open annulus { z C : R 1 < | z z 0 | < R 2 } . Furthermore, if C r ( t ) = z 0 + re it , with R 1 < r < R 2 , 0 t 2 π , then a n = 1 2 πi C r f ( z ) ( z z 0 ) n +1 dz, b n = 1 2 πi C r f ( z )( z z 0 ) n - 1 dz. Proof. Omitted. More compactly: f ( z ) = n = -∞ c n ( z z 0 ) n , which converges in the open annulus { z C : R 1 < | z z 0 | < R 2 } and, for all n Z , c n = 1 2 πi C r f ( z ) ( z z 0 ) n +1 dz. The above series is called the Laurent series of f ( z ) about z 0 and also the Laurent expansion of f ( z ). The series - 1 n = -∞ c n ( z z 0 ) n is called the principal part of the Laurent series.
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