Cauchy’s Integral Theorem states that if a function is holomorphic (complex differentiable) on and inside a simple closed contour, then the integral of that function over the contour is zero. This theorem is fundamental in complex analysis and implies that the integral of a holomorphic function depends only on the values of the function inside the contour, not on the path taken.
The Residue Theorem provides a powerful method for evaluating complex integrals. It states that if a function is meromorphic (holomorphic except for isolated poles) within a region bounded by a simple closed contour, the integral of the function over that contour is equal to \(2\pi i\) times the sum of the residues of the function at its poles inside the contour. This theorem allows for the computation of integrals that may be difficult to evaluate using other methods.
The primary difference lies in their application: Cauchy’s Integral Theorem applies to holomorphic functions over closed contours, resulting in a zero integral, while the Residue Theorem applies to meromorphic functions with poles, allowing for the evaluation of integrals by summing the residues at those poles. In essence, Cauchy’s theorem is a special case of the Residue Theorem when there are no poles inside the contour.
Cauchy’s Integral Theorem is used when you want to demonstrate that the integral of a holomorphic function over a closed contour is zero, often in contexts where the function is known to be holomorphic everywhere inside the contour. It is particularly useful for establishing properties of holomorphic functions and their integrals without needing to consider singularities.
Certainly! To apply the Residue Theorem, consider the integral of a function like \(f(z) = \frac{1}{z^2 + 1}\) over a closed contour that encloses the poles at \(z = i\) and \(z = -i\). You would first calculate the residues at these poles. The residue at \(z = i\) is \(\frac{1}{2i}\), and at \(z = -i\) it is \(-\frac{1}{2i}\). The integral over the contour would then be \(2\pi i\) times the sum of these residues, leading to the evaluation of the integral directly without computing it along the contour itself.