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