Page 48 - Calculus with Complex Numbers
P. 48
Ffgure 4.5
Observe that the orientation of p , y4 is given by decreasing t. W e indicate this by
writing 1 k: t k: 0 instead of 0 :jq t :jq 1.
4 .3 C ontou r i ntegrals
Given a contour y and a function flz) defined for z eE y we deline
m
/
j .t'(z) dz = zyti-xlj )-) .f(z) dz.
Theorenls 1 and 2 of S ection 4.1 renlaùA valid in the conlplex context also the
combination rules for integrals. The inequalities generalise to the following.
4.4 Estim ate Iem m a
lf I.f(z)I:i M for z e y, then
flz) dz :% M lv,
/
r
where l is the length of y .
Regarding evaluation of contour integrals we give three methods.
4 .5 M ethod I : Substituting the param etri c fun ction
W e describe the method by way of examples.
Example 1 Hvaluate
n dz,
Z
p'
where y is the unit circle parametrised by letting z = eit where 0 :jq t :jq 2zr .
