Page 17 - Calculus with Complex Numbers

Ayplication 4 Formulae of the above type are useful for integrating powers of
cos 0 sin 0. For example

j coszp vlo - j . (1 + coscp) vlo - .) (é? + Sinzpj ,
)
1
j sinz ovlo - j ) (1 - coscp) vlo - ) (é? - Sinzpj .
1

I .9 rlth roots
Suppose we have two complex numbers z = reio w = seis . lf we multiply them
together we obtain

i (9 +/)
.:: N7 = rs c ,

which shows that Izv?l= IzII 'r1as claimed in Section 1.2. Also that arg zw =

arg c + arg w. Inparticular, taking z = w we have .:2 = rlelio and more generally
cn = rnenio It follows that
1/n
C 1/n ioln

' = r ' d ' .
Observe that rl/n is the unique positive real rlth root of r whilst eio/n has n
,
possible values.
For example, if z = - 8 then we have

i:c 3fn' 5i:c
.1 = 8c = 8c = 8c = . . .
1 /3 tz i:c(3 g i n' g 5f n'/3
C = t ty , ty .
y
,
Hven though arg (-8) has infinitely many values, there are only 3 distinct cube
roots. We define the principal value of (- 8)1/3 to be that which corresponds to the
principal value of arg (-8), namely n'. So (-8)1/3 = lei=I3 (PV).