Mathematical Methods for Physicists: A concise introduction - Site Map

COMPLEX NUMBERS
p! 10 !
1 ‡ i 3
p
ˆ
10ˆ
2ei=3
e
10ˆ 2i=3 e
20i=3
1 i 3 2e i=3
ˆ e 6i e 2i=3 ˆ 1cos…2=3†‡i ‰ sin…2=3† Š ˆ 1 p
2 ‡ i 3
2 :
De Moivre's theorem and roots of complex numbers
If z 1 ˆ r 1 e i 1
and z 2 ˆ r 2 e i 2
, then
z 1 z 2 ˆ r 1 r 2 e i… 1‡ 2† ˆ r 1 r 2 ‰cos… 1 ‡ 2 †‡i sin… 1 ‡ 2 †Š:
A generalization of this leads to
z 1 z 2 z n ˆ r 1 r 2 r n e i… 1‡ 2 ‡‡ n †
ˆ r 1 r 2 r n ‰cos… 1 ‡ 2 ‡‡ n †‡i sin… 1 ‡ 2 ‡‡ n †Š;
if z 1 ˆ z 2 ˆˆz n ˆ z this becomes
from which it follows that
z n ˆ…re i † n ˆ r n ‰cos…n†‡i sin…n†Š;
…cos ‡ i sin † n ˆ cos…n†‡i sin…n†;
…6:4†
a result known as De Moivre's theorem. Thus we now have a general rule for
calculating the nth power of a complex number z. We ®rst write z in polar form
z ˆ r…cos ‡ i sin †, then
z n ˆ r n …cos ‡ i sin † n ˆ r n ‰cos n ‡ i sin nŠ:
…6:5†
The general rule for calculating the nth root of a complex number can now be
derived without diculty. A number w is called an nth root of a complex number
z if w n ˆ z, and we write w ˆ z 1=n .Ifz ˆ r…cos ‡ i sin †, then the complex number
w 0 ˆ
np
r
cos
n ‡ i sin n
is de®nitely the nth root of z because w n 0 ˆ z. But the numbers
w k ˆ
np ‡ 2k
r cos ‡ i sin ‡ 2k
; k ˆ 1; 2; ...; …n 1†;
n
n
are also nth roots of z because w n k ˆ z. Thus the general rule for calculating the nth
root of a complex number is
w ˆ
np ‡ 2k
r cos ‡ i sin ‡ 2k
; k ˆ 0; 1; 2; ...; …n 1†: …6:6†
n
n
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