Residue theorem

Residue Theorem: The Residue theorem states that the value of a complex function evaluated at a point within the circle of convergence of that function...

Residue Theorem:

The Residue theorem states that the value of a complex function evaluated at a point within the circle of convergence of that function is equal to the value of that function at the point itself.

Specifically, the residue theorem states:

If (f(z)) is a complex function, (f(z)) has a finite number of isolated zeros within the circle (C_R(a)), and (f(z)) is continuous on (C_R(a)), then:

$\lim_{z\to a} (z-a)f(z) = f(a)$

Intuitively, this theorem says:

If you graph the complex function on a circle centered at the origin, and then draw a circle around the origin that is just touching the curve of the function, the value of the function at the origin will be equal to the value of the function at any point on the outer circle.

Examples:

(f(z) = \frac{1}{z}) has a removable zero at (z = 0), so (f(0) = 1).
(f(z) = \frac{1}{z^2}) has a pole at (z = 0), so (f(0) = 0).
(f(z) = \sin(z)) has no zeros, so (f(a) = 0) for any (a).

Note:

The residue theorem is a powerful tool for evaluating complex function values. It can be used to evaluate functions at points where it would be difficult or impossible to use other methods, such as direct substitution