University Mathematics Handbook (2015)

XII. Complex Functions

Chapter 9. Residue and its Applications

9.1  Definition

a.  Coefficient in an expansion of into a Laurent series in ring is called a residue of on . It is denoted .

b.   when is a closed curve surrounding and is entirely in that ring.

9.2  Calculating Residues

a.  If is a simple pole of , then

b.  If point is a pole of order of , then

.

c.  Residue theorem: Let be an analytic function in domain except a finite number of singular points , then, for every closed curve surrounding these points, and is in , there holds