Complex Integration and Residue Theorem The Complex Integration Theorem states that the Riemann integral of a function defined on a closed contour in...
The Complex Integration Theorem states that the Riemann integral of a function defined on a closed contour in the complex plane can be calculated by evaluating the Residue at infinity. This means that the integral can be calculated directly by examining the behavior of the function as it approaches infinity.
Specifically, the theorem states that:
If f(z) is analytic within and on the contour C, and C is closed, then:
Where:
C is the contour.
f(z) is the function to be integrated.
∞ is the infinite limit of integration.
Furthermore, the Residue theorem provides a way to calculate the value of the integral by evaluating the following limit:
The Residue theorem can be applied in various ways to evaluate definite integrals involving complex functions. This allows us to calculate the areas of complex regions, evaluate the real part of a complex function, and solve certain types of improper integrals.
Examples:
\begin{split} \int_C \frac{1}{z} dz &= \left[ z^{-1} \right]_0^1 \\\ & = \ln(1) \\\ & = \infty \end{split}
This means the integral diverges.
\begin{split} \lim_{z \to \infty} \left(z^2 \right) f(z) &= \lim_{z \to \infty} z^2 \left(\frac{1}{z}\right) \\\ & = 1 \\\ & \neq 0 \end{split}
This shows that the integral does not converge