Page 50 - Calculus with Complex Numbers

4 .6 M ethod 2: U si ng the fundam e ntal theore m of caI cuI us
lf the contour y has end points a b with orientation a to b and if the function
f (z) has a primitive F (z) on y (F/ (z) = f (z)), then

(1 + ï)S
3

M ore generally we have the following theorem .

Theorem 3 lf y is any closed contotlr and if f (z) has a primitive on y then

flz) dz = 0.
;'
Corollal'y 1 (See Example 1 of Section 4.5) lf y is the unit circle, then for all
n # - 1 we have

z n dz = 0.
r
Proof For n # - 1 the function zn has the primitive znh'î/ (rl + 1) on y .
Corollal'y 2 The function 1/z has no primitive on the unit circle.

Proof We showed in Section 4.5 that

lt might be thought that log z is a primitive for 1/z on the unit circle. However
by Theorem 1 of Chapter 3, any F(z) such that F/ (z) = f (z) must be contin-
uous. W hichever values we take for log z on the unit circle there is bound to be
a discontinuity. For example log z (PV) has a discontinuity at z = - 1.