The key word linked to contour integration is “analyticity” or the absense thereof:
|
f(z) ={ \mathop{∑
}}_{n=0}^{∞}{f}^{(n)}(c){{(z − c)}^{n}\over
n!}
| (A.1) |
exists for every point c
inside R.
In most cases we are actually interested in functions that are not analytic; if this only happens at isolated points
(i.e., we don’t consider a ”line of singularities”, usually called a ”cut” or ”branch-cut”) we can still expand the
function in a Laurent series
|
f(z) ={ \mathop{∑
}}_{n=−∞}^{∞}{f}^{(n)}(c){{(z − c)}^{n}\over
n!}
| (A.2) |
How we obtain the coefficients {f}^{(n)}(c) from the function is closely linked to the problem of contour integration.