Paths and contour integrals
Definition. A path is a piecewise \(C^1\) map \(\gamma:[a,b]\to U\). For a continuous function \(f\), define \[ \int_\gamma f(z)\,\mathrm{d}z=\int_a^b f(\gamma(t))\gamma'(t)\,\mathrm{d}t. \] Reparametrization preserving orientation does not change the integral.
Length estimate
Proposition. If \(M=\sup_{z\in\gamma}|f(z)|\) and \(L(\gamma)\) is the path length, then \(|\int_\gamma f\,\mathrm{d}z|\le M L(\gamma)\). This estimate is the basic analytic control behind limiting arguments with contours.
Primitives
Theorem. If \(f=F'\) on \(U\), then \(\int_\gamma f\,\mathrm{d}z=F(\gamma(b))-F(\gamma(a))\). Hence every closed contour integral of a function with a primitive is zero.
Path independence
Corollary. On a connected open set, a continuous function has a primitive if and only if its integral over every closed path is zero. One constructs the primitive by fixing a base point and integrating along paths.
Integral of reciprocal powers
Calculation. On the circle \(\gamma(t)=a+re^{it}\), \(0\le t\le2\pi\), one has \(\int_\gamma (z-a)^n\,\mathrm{d}z=0\) for \(n\ne-1\), while \(\int_\gamma (z-a)^{-1}\,\mathrm{d}z=2\pi i\). This single exceptional exponent is the source of residues.
Orientation
Remark. Reversing a contour changes the sign of the integral. Positively oriented boundary curves keep the region on the left as they are traversed.
Piecewise smooth and rectifiable paths
For a piecewise \(C^1\) path \(\gamma:[a,b]\to\Omega\), \[ \int_\gamma f(z)\,\mathrm{d}z =\int_a^b f(\gamma(t))\gamma'(t)\,\mathrm{d}t. \] The definition extends to rectifiable paths using line integration against the complex measure induced by \(\gamma\). It is invariant under orientation- preserving piecewise \(C^1\) reparametrization and changes sign under reversal.
Proposition (ML estimate). If \(|f|\le M\) on the image of \(\gamma\), then \[ \left|\int_\gamma f(z)\,\mathrm{d}z\right| \le M\,L(\gamma), \qquad L(\gamma)=\int_a^b|\gamma'(t)|\,\mathrm{d}t. \]
Proof. Use the triangle inequality for the real integral: \[ \left|\int_a^bf(\gamma(t))\gamma'(t)\,\mathrm{d}t\right| \le\int_a^b|f(\gamma(t))|\,|\gamma'(t)|\,\mathrm{d}t. \]
Primitives and path independence
Theorem. For continuous \(f:\Omega\to\mathbb C\), the following are equivalent on a connected open set:
- \(f\) has a holomorphic primitive \(F\), \(F'=f\);
- the integral of \(f\) over every closed piecewise \(C^1\) path is zero;
- \(\int_\gamma f\,\mathrm{d}z\) depends only on the endpoints.
Proof. If \(F'=f\), the chain rule gives \(\int_\gamma f=F(\gamma(b))-F(\gamma(a))\). Endpoint dependence immediately implies zero closed integrals. Conversely, fix \(z_0\) and define \(F(z)=\int_{z_0}^zf\,\mathrm{d}z\) along any path. Closed-integral vanishing makes this well defined. On a small disc, compare \(F(z+h)-F(z)\) along the line segment and use continuity: \[ \frac{F(z+h)-F(z)}h =\int_0^1f(z+th)\,\mathrm{d}t\to f(z). \]
The index of a closed curve
Definition. If \(a\notin\gamma\), the winding number is \[ \operatorname{Ind}(\gamma,a) =\frac1{2\pi i}\int_\gamma\frac{\mathrm{d}z}{z-a}. \]
Proposition. \(\operatorname{Ind}(\gamma,a)\in\mathbb Z\), is locally constant on \(\mathbb C\setminus\gamma\), and is zero on the unbounded component.
Proof. Along the path choose a continuous argument locally and lift \((\gamma(t)-a)/|\gamma(t)-a|\) to \(e^{i\theta(t)}\). Closedness implies \(\theta(b)-\theta(a)=2\pi k\). Integrating the logarithmic derivative gives the index \(k\). Continuity in \(a\) and integer-valuedness imply local constancy; the ML estimate forces the integral to zero as \(|a|\to\infty\).