A contour integral is a path integral ∫_C f(z) dz of a function f in the complex plane along an oriented, usually closed curve C. Such integrals can be evaluated using Cauchy’s Residue Theorem; using appropriately chosen contours and integrands, and then taking real or imaginary parts, a vast range of integrals and series can be determined. For example, the substitution z = e^iθ‎ changes the integral ${\displaystyle {\int }_0^{2\pi }\frac{\text{d}\theta }{{\left(2+\text{cos}\theta \right)}^2}}$ into the path integral ${\displaystyle {\int }_C^{}\frac{-4iz\text{d}z}{{\left(z^2+4z+1\right)}^2}}$ where C is the positively oriented unit circle centred at 0. The new integrand has a double pole inside C, the residue of which can be determined and so the integral evaluated.