3: Complex integration and Cauchy's theorem

Complex Integration and Cauchy's Theorem

Complex integration, extending the concept of integration to functions of a complex variable, is defined along contours (paths) in the complex plane. A contour is a piecewise-smooth curve $C$, parameterized by $z(t) = x(t) + iy(t)$ for $a \leq t \leq b$. The contour integral of a complex function $f(z)$ along $C$ is:

$$\int_C f(z) dz = \int_a^b f(z(t)) z'(t) dt.$$

This integral decomposes into real line integrals involving the real and imaginary parts of $f(z) = u(x,y) + iv(x,y)$ and $dz = dx + i dy$.

Unlike real integrals, the value of a complex contour integral depends critically on the path taken between two points. However, a fundamental result emerges when integrating analytic functions. A function is analytic (holomorphic) in a domain $D$ if it is differentiable at every point in $D$, satisfying the Cauchy-Riemann equations.

Cauchy's Theorem (also called the Cauchy-Goursat Theorem) provides the cornerstone:
If $f(z)$ is analytic at all points on and inside a simple closed contour $C$ (a loop that doesn't intersect itself), then

$$\oint_C f(z) dz = 0.$$

This profound result has immediate, crucial consequences:

1. Path Independence: If $f(z)$ is analytic in a simply connected domain $D$ (a region with no "holes"), the integral $\int_C f(z) dz$ depends only on the endpoints of $C$, not the specific path connecting them within $D$. For any two paths $C_1$ and $C_2$ in $D$ sharing the same start and end points:

   $$\int_{C_1} f(z) dz = \int_{C_2} f(z) dz.$$

   This allows the definition of an antiderivative $F(z)$ such that $F'(z) = f(z)$ in $D$, analogous to the Fundamental Theorem of Calculus.

2. Deformation of Contour: If $f(z)$ is analytic in a region bounded by two simple closed contours $C_1$ and $C_2$ (where $C_2$ lies entirely inside $C_1$), and analytic in the region between them, then:

   $$\oint_{C_1} f(z) dz = \oint_{C_2} f(z) dz.$$

   The integral around the outer contour equals the integral around the inner contour. This principle allows complex contours to be deformed continuously within the region of analyticity without changing the integral's value.

Cauchy's Theorem highlights the stringent constraints analyticity imposes. The integral of an analytic function around a closed loop vanishes, while even minor singularities (points where analyticity fails) within the loop drastically alter the integral's value (a key idea later exploited by the Residue Theorem). Understanding these properties is essential for evaluating complex integrals and underpins much of complex analysis.