List of basic formulas related to the integral over a closed contour:
- Integral over a closed contour (Cauchy’s theorem):
[
\oint_C f(z) \, dz = 2\pi i \sum \text{Residues of } f \text{ inside } C
]
where ( C ) is a closed contour and ( f(z) ) is an analytic function. - Cauchy’s formula for the derivative:
[
f^{(n)}(a) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(za)^{n+1}} \, dz
]
where ( a ) is a point inside the contour ( C ), and ( n ) is the order of the derivative. - Residue theorem:
[
\oint_C f(z) \, dz = 2\pi i \sum_{k} \text{Res}(f, z_k)
]
where ( z_k ) are the poles of the function ( f(z) ) inside the contour ( C ). - Integral over a closed contour for a single-valued function:
If ( f(z) ) is analytic in the domain containing the closed contour ( C ), then:
[
\oint_C f(z) \, dz = 0
]