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THE INTEGRAL OF A COMPLEX
FUNCTION CAUCHY’S THEOREM
CHAPTER 20 CONSEQUENCES OF CAUCHY’S
THEOREM
Complex

Integration

20.1 The Integral of a Complex Function

Real-valued functions are integrated over intervals. Complex functions are integrated over curves
and have many properties in common with line integrals of vector fields. The notions of contin-
uous, differentiable, smooth, and piecewise smooth curves were developed in Section 12.1. Here
we will use complex notation and represent points (x, y) on a curve as complex numbers x + iy.

EXAMPLE 20.1
it
Let γ(t) = e for 0 ≤ t ≤ 3π/2. Then γ is a simple, smooth curve with initial point γ(0) = 1 and
terminal point γ(3π/2) =−i. γ is therefore not closed. Since γ(t) = cos(t) + i sin(t), this curve
has coordinate functions x = cos(t) and y = sin(t). The graph of γ is the three-quarter circle of
Figure 20.1.

y

x
1

–i

FIGURE 20.1 γ in Exam-
ple 20.1.

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