214 CHAPTER 6 • COMPLEX lNTECRATION- Let z (t) = x (t) + iy (t), for a :$ t :$ b, be a smooth curve. Give a meaning for
each of the following expressions.
(a) z' (t).
(b) lz' (t)I dt.
(c) J: z' (t) dt.(d) J: lz' (t)I dt.
17. Evaluate f c cos z dz, where C is the polygonal path from 0 to 1 + i that consists
of the line segments from 0 to 1 and 1 to 1 + i.
18. Let f (t) = eu be defined on a :$ t :$ b, where a = 0, and b = 2?T. Show that
there is no number c E (a, b) such that f (c) (b -a) = J: f (t) dt. In other words,
the mean value theorem for definite integrals that you learned in calculus does not
hold for complex functions.- Use the ML inequality to show that IPn (x)I :$ 1, where Pn is the nth Legendre
polynomial defined on -1 :$ x :$ 1 by Pn (x) = ~Ion (x + iVI='X2 cos of dO. - Explain how contour integrals in complex analysis and line integrals in calculus
are different. How are they similar?
6.3 The Cauchy- Goursat Theorem
The Cauchy-G-Oursat theorem states that within certain domains the integral of
an analytic function over a simple closed contour is zero. An extension of this
theorem allows us to replace integrals over certain complicated conto~rs with
integrals over contours that are easy to evaluate. We demonstrate how to use
the technique of partial fractions with the Cauchy- Goursat theorem to evaluate
certain integrals. In Section 6.4 we show that the Cauchy- Goursat tli.eorem
implies that an analytic function has an antiderivative. To begin, we need to
introduce some new concepts.
Recall from Section 1.6 that each simple closed contour C divides the plane
into two domains. One domain is bounded and is called the interior of C;
the other domain is unbounded and is called the exterior of C. Figure 6.15
illustrates this concept, which is known as the Jordan curve theorem.
Recall also that a domain D is a connected open set. In particular, if z 1
and z2 are any pair of points in D, then they can be joined by a curve that lies
entirely in D. A domain D is said to be a simply connected domain if the
interior of any simple closed contour C contained in D is contained in D. In
other words, there axe no "holes" in a simply connected domain. A domain that
is not simply connected is said to be a multiply connected domain. Figure
6.16 illustrates uses of the terms simply connected and multiply connected.
Let the simple closed contour C have the parametrization C: z (t) = x (t) +
iy ( t) for a $ t $ b. Recall that if C is parametrized so that the interior of
C is kept on the left as z (t) moves around C, then we say that C is oriented