6.2 • CONTOURS ANO CONTOUR INTEGRALS 211yFigure 6.10 The distances lz -i i and lz +ii for z on C.
- EXAMPLE 6.11 Use Inequality (6-23) to show that
1 1z^2 ~ 1 dzl ~ 2~'
where C is the straight-line segment from 2 to 2 + i.
Solution Here lz^2 + ll = lz -ii lz +ii, and the terms lz - ii and lz + ii repre-
sent the distance from the point z to the points i and -i, respectively. Referring
to Figure 6.10 and using a geometric argument, we get
lz - i i ~ 2 and lz +ii ~ J5, for z on C.
Thus, we have
1 ll/(z)I = lz-il lz+il ~ 2VS = M.
Because L, the length of C , equals 1, Inequality (6- 23 ) implies thatIL z2 ~ 1 dzl '!,ML= 2~·
- ------~EXERCISES FOR SECTION 6.2
- Give a parametrization of each contour.
(a) C = C 1 + C2, as indicated in Figure 6.11.