1

Complex Variables 1.4 Integer and Fractional Powers of Complex

Numbers

In general: i zi = i ri (1.3-16) ii

So: zn = rncis(n)= rn = rn[cos(n)+isin(n)] when n n

If the power is an integer:

De Moivre’s Theorem

mimr

mm

mim

rmimrz

z

m

mm

m

m

sincos

sincos

sincos1

sincos

1122

nini

ninrirz

n

nnnn

sincossincos

sincossincos

2

Complex Variables

If the power is a fractional: z1/m

Let:

Rising that to the m-power

So:

And

Finally:

k = 0,1,2,3,…..m-1, m, m+1 …

.....3,2,1,0,21

2

kkm

km

sincos ir

3

Complex Variables

k = 0,1,2,3,…..m-1, m, m+1 …

but

So:

k = 0,1,2,3,…..m-1; m 1 - m-fold valued root

The sum of all the roots is 0 !!

1

22

212;02

2

k

mmm

mk

m

m

02

sin2

cos

02

sin2

cos2

sin2

cos

1

0

1

0

11

0

1

m

k

m

k

m

m

k

m

m

k

mi

m

k

m

m

k

mi

m

k

mr

m

k

mi

m

k

mr

4

Complex Variables

w1+ w2+ w3= 0

y

x

-i

i

-1 1

EXAMPLE

Find the three cube roots of 1.

r = 1, arg z = 0, m = 3

So:

2

3

2

1

3

4sin

3

4cos1

3

22

3

012

2

3

2

1

3

2sin

3

2cos1

3

12

3

011

10sin0cos13

010

3

2

1

iiciswk

iiciswk

iciswk

5

Complex Variables

zn = rn[cos(n) +i sin(n)]

6

Complex Variables

)134.1(1,.....,3,2,1,02

sin2

cos

mk

m

nk

m

ni

m

nk

m

nrz mnmn