chapter-6

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Laurent Series and Isolated Singularities
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2. Isolated Singularities of an Analytic Function
Definition 1 A point z0 is an isolated singularity of f (z ) if f (z ) is analytic in some punctured disk 0 < |z − z0 | < r centered at z0 . Example. 1 1 1 , , 1 z sin z sin z
1. The Laurent Decomposition
Example. Find all possible Laurent expansions centered at z = 0 of 1 the function f (z ) = (z −1)( z −2) . Find the Laurent expansion centered at z = −1 that converges at z = 3/2 and determine the largest open set on which this series converges.
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Laurent Series and Isolated Singularities
May 11, 2014
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2. Isolated Singularities of an Analytic Function
Definition 1 A point z0 is an isolated singularity of f (z ) if f (z ) is analytic in some punctured disk 0 < |z − z0 | < r centered at z0 . Example. 1 1 1 , , 1 z sin z sin z
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Laurent Series and Isolated Singularities
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Pole Definition 3 The isolated singularity of f (z ) at z0 is defined to be a pole if there is N > 0 such that a−N = 0 but ak = 0 for all k < −N .
Theorem
Let z0 be an isolated singularity of f (z ). Then z0 is a pole of f (z ) of order N if only if f (z ) = g (z )/(z − z0 )N , where g (z ) is analytic at z0 and g (z0 ) = 0.
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Laurent Series and Isolated Singularities
Laurent Series and Isolated Singularities
May 11, 2014
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Laurent Series and Isolated Singularities
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1. The Laurent Decomposition
Theorem
Laurent Decomposition Suppose 0 ≤ ρ < σ ≤ +∞, and suppose f (z ) is analytic for ρ < |z − z0 | < σ . Then f (z ) can be decomposed as a sum f (z ) = f0 (z ) + f1 (z ), where f0 (z ) is analytic for |z − z0 | < σ , and f1 (z ) is analytic for |z − z0 | > ρ and at ∞. If we normalize the decomposition so that f1 (∞) = 0, then the decomposition is unique.
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Laurent Series and Isolated Singularities
May 11, 2014
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1. The Laurent Decomposition
Example. Find all possible Laurent expansions centered at z = 0 of 1 the function f (z ) = (z −1)( z −2) . Find the Laurent expansion centered at z = −1 that converges at z = 3/2 and determine the largest open set on which this series converges.
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Laurent Series and Isolated Singularities
May 11, 2014
5 / 13
2. Isolated Singularities of an Analytic Function
Definition 1 A point z0 is an isolated singularity of f (z ) if f (z ) is analytic in some punctured disk 0 < |z − z0 | < r centered at z0 . Example. 1 1 1 , , 1 z sin z sin z
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Laurent Series and Isolated Singularities
May 11, 2014
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2. Isolated Singularitien 1 A point z0 is an isolated singularity of f (z ) if f (z ) is analytic in some punctured disk 0 < |z − z0 | < r centered at z0 . Example. 1 1 1 , , 1 z sin z sin z
k =+∞ k =−∞ ak (z
− z0 )k
that converges absolutely at each point of the annulus, and that converges uniformly on each subannulus r ≤ |z − z0 | ≤ s, where ρ < r < s < σ . The coefficients are uniquely determined by f (z ), and they are given by ak =
1 2π i f (z ) |z −z0 |=r (z −z0 )k +1 dz ,
k = 0, ±1, ±2, · · · .
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for any fixed r , ρ < Laurent r <Series σ . and Isolated Singularities ()
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Laurent Series and Isolated Singularities
May 11, 2014
5 / 13
2. Isolated Singularities of an Analytic Function
Definition 1 A point z0 is an isolated singularity of f (z ) if f (z ) is analytic in some punctured disk 0 < |z − z0 | < r centered at z0 . Example. 1 1 1 , , 1 z sin z sin z
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Laurent Series and Isolated Singularities
May 11, 2014
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Removable singularity Definition 2 The isolated singularity of f (z ) at z0 is defined to be a removable singularity if ak = 0 for all k < 0.
() Laurent Series and Isolated Singularities May 11, 2014 2 / 13
1. The Laurent Decomposition
Theorem
Laurent Series Expansion Suppose 0 ≤ ρ < σ ≤ +∞, and suppose f (z ) is analytic for ρ < |z − z0 | < σ . Then f (z ) has a Laurent expansion
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Laurent Series and Isolated Singularities
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Removable singularity Definition 2 The isolated singularity of f (z ) at z0 is defined to be a removable singularity if ak = 0 for all k < 0.
Theorem