1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

t(b)

• ~

(a) A curve that is simple.

z(b)

z~

(c) A curve that is not simple

and not closed.

6.2 • CONTOURS AND CONTOUR INTEGRALS 199

z(a) = z(b)

(b) A simple close<! curve.

z(a) = t(b)

( d) A closed curve that is not simple.

Figure 6.1 The terms simple and closed used to describe curves.

one-sided derivatives^1 of x (t) and y (t) to exist at the endpoints of the interval.
As in Section 6.1, the derivative z' is

z' (t) = x' (t) + iy' (t), for a~ t ~ b.

The curve C defined by Equation,{6-10) is said to be a smo o t h curve if z'
is continuous and nonzero on the interval. If C is a smooth curve, then C has a
nonzero tangent vector at each point z (t), which is given by the vector z ' (t). If
x' (to)= 0, then the tangent vector z' (to)= iy' (to) is vertical. If x' (to) :f: 0,
then the slope ~ of the tangent line to C a.t the point z (to) is given by ~ '.~~~l ·
Hence for a smooth curve the angle of inclination() (t) of its tangent vector z ' (t)
is defined for all values oft E [a, b] and is continuous. Thus, a smooth curve has
no corners or cusps. Figure 6.2 illustrates this concept.

t(t) z'(t) }\____ z(t

0--3' J ~)

t(a) t(a)

(a) A smooth curve. (b) A curve that is not smooth.

Figure 6.2 The term smooth used to describe curves.

(^1) The derivatives on the right, x' (a+), and on the left, x' (&-),are defined by the limits
I ( +) - I' X (t) -X (a) d / (b-) - I' X (t) - X (b)
x a - t.!~+ t -a an x - t.!~- t - b ·