Complex Analysis 17 | Complex Integration on Real Intervals [dark version]

Complex integration on real intervals sets up the machinery for integrating complex-valued functions by reducing everything to ordinary real Riemann integration. The core move is to define an integral of a complex function along a continuous map from a real interval [A, B] into the complex plane: if γ is continuous and complex-valued, then the complex integral is built from the real and imaginary parts separately. Concretely, the integral over γ is computed as the real Riemann integral of Re(γ) plus i times the real Riemann integral of Im(γ). This makes the definition well-posed for continuous γ and preserves familiar algebraic rules like linearity.

A key limitation also emerges immediately: unlike real integrals, complex numbers have no natural order, so “monotonicity” cannot carry over. Instead, the course replaces it with an inequality controlled by absolute values. The central estimate is that the magnitude of a complex integral is bounded by the integral of the magnitude of the integrand: |∫_A^B γ(t) dt| ≤ ∫_A^B |γ(t)| dt. This estimate becomes a foundational tool for later results, especially once integration moves from real intervals to arbitrary curves in the complex plane.

To make the definition feel concrete, the transcript works through an example using the unit circle. Take γ(t) = e^{it} on t ∈ [A, B]. Splitting into components gives Re(e^{it}) = cos t and Im(e^{it}) = sin t, so the complex integral becomes ∫_A^B cos t dt + i∫_A^B sin t dt. Using the fundamental theorem of calculus, the antiderivatives are sin t and −cos t, respectively. After simplifying, the result collapses back into a compact exponential form: ∫_A^B e^{it} dt = (1/i)(e^{iB} − e^{iA}), matching the idea that the fundamental theorem of calculus works in this complex setting just as it does for real integrals.

The proof of the absolute-value estimate is where the method really matters. The argument introduces w = ∫_A^B γ(t) dt and normalizes it by its magnitude: if w ≠ 0, define C = w/|w|, which lies on the unit circle so |C| = 1. Multiplying the integral by C^{-1} turns the complex number w into a positive real number |w|, allowing the real-part machinery to take over. Linearity lets C^{-1} move inside the integral, and then the inequality |Re(z)| ≤ |z| for any complex z bounds the integrand’s real part by its modulus. Monotonicity of the ordinary real Riemann integral applies to this real inequality, yielding |∫_A^B γ(t) dt| ≤ ∫_A^B |γ(t)| dt. The equality case aligns with the normalization choice.

With this real-interval complex integral in place—and with the absolute-value bound established—many familiar calculus tools are expected to transfer from the real Riemann setting: the fundamental theorem of calculus, substitution, and integration by parts. The next step is to extend these ideas from real intervals to curve integrals along general paths in the complex plane (the contour/line integral framework).