Complex contour integral

Definition

Let $\gamma:[a,b]\to\mathbb C$ be piecewise $C^1$ , and let $f$ be continuous on the image of $\gamma$ . The complex contour integral of $f$ along $\gamma$ is

\int_\gamma f(z)\,dz =\int_a^b f(\gamma(t))\gamma'(t)\,dt,

with the ordinary integral taken separately in real and imaginary parts on each smooth piece.

Invariance and orientation

The integral is invariant under orientation-preserving piecewise $C^1$ reparametrization. Reversing orientation changes its sign, and concatenating compatible curves adds their integrals. The value depends on the parametrized oriented contour, not merely on its image.

Basic estimate

If $L(\gamma)$ is the length of the contour, then

\left|\int_\gamma f(z)\,dz\right| \le L(\gamma)\max_{z\in\gamma([a,b])}|f(z)|.

This estimate underlies convergence arguments and the derivative estimates obtained from the Cauchy integral formula.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter IV, §1.