6.4 • THE FUNDAMENTAL THEOREMS OF INTEGRATION 231yFigure 6.31 The contours r i and r 2 and the line segment r.
R em ark 6.2 It is important to stress that the line integral of an analytic
function is independent of path. In Example 6.9 we showed that f c, z dz =
fc. z dz = 4 + 2i, where Ci and C2 were different contours joining - 1 - i to
3 + i. Because the integrand f (z) = z is an analytic function, Theorem 6.8 lets
us know ahead of time that the value of the two integrals is the same; hence one
calculation would have sufficed. If you ever have to compute a line integral of
an analytic function over a difficult contour, change t he contour to something
easier. You are guaranteed to get the same answer. Of course, you must be sure
that the function you're dealing with is analytic in a simply connected domain
containing your original and new contours. •
If we set z = z1 in Theorem 6.8, then we obtain the following familiar result
for evaluating a definite integral of an analytic function.