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SINGULARITIES THE RESIDUE THEOREM
EVALUATION OF REAL INTEGRALS
CHAPTER 22 RESIDUES AND THE INVERSE LAPLACE
TRANSFORM
Singularities

and the Residue

Theorem

22.1 Singularities

We will use the Laurent expansion to classify points at which complex functions are not
differentiable.

We say that f has an isolated singularity at z 0 if f is differentiable in an annulus 0 <
|z − z 0 | < R, but not at z 0 itself.

For example, 1/z has an isolated singularity at z = 0, and sin(z)/(z − π) has an isolated
singularity at z = π.
We will identify three different kinds of isolated singularities, depending on the coefficients
in the Laurent expansion.

Classification of Singularities

Suppose f has an isolated singularity at z 0 . Let the Laurent expansion of f (z) in an annulus
0 < |z − z 0 | < R be
∞

n
f (z) = c n (z − z 0 ) .
n=−∞
1. z 0 is a removable singularity if c n = 0 for every negative integer n.

729

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October 14, 2010 15:37 THM/NEIL Page-729 27410_22_ch22_p729-750