An integral obtained by contour integration. The particular path in the complex plane used to compute the integral is called a contour.
As a result of a truly amazing property of holomorphic functions, a closed contour integral can be computed simply by summing the values of the complex residues inside the contour.
Watson (1966 p. 20) uses the notation 
to denote the contour integral of 
with contour encircling the
point 
once in a counterclockwise direction.
Renteln and Dundes (2005) give the following (bad) mathematical joke about contour integrals:
Q: What's the value of a contour integral around Western Europe? A: Zero, because all the Poles are in Eastern Europe.
