6 .4 • THE FUNDAMENTAL THEOREMS OF INTEGRATION 229- Evaluate fct(o) lz l^2 expz dz.
12. Suppose t hat f (z) = u (r , 9) +iv (r, 9) is analytic for all values of z = re^19 • Show
that
{2"
lo [u (r,9) cos 9 - v (r ,9) sin 9) d9 = 0.Hint: Integrate f around the circle C{ (0).- If C is the figure eight contour shown in Figure 6.28(a).
(a) evaluate f c (z^2 - z)-^1 dz.(b) evaluate f c (Zz - 1) (z^2 - z)-^1 dz.- Compare the various methods for evaluating contour integrals. What are the
limitations of each method?
6.4 The F\mdamental Theorems of Integration
OF INTEGRATION
Let f be analytic in the simply connected domain D. The theorems in this
section show that an antiderivative F can be constructed by contour integration.
A consequence will be the fact that in a simply connected domain, the integral
of an analytic function f along any contour joining z 1 to z2 is the same, and
its value is given by F (z 2 ) - F (z 1 ). As a result , we can use the antiderivative
formulas from calculus to compute the value of definite integrals.