Mathematical Methods for Physicists: A concise introduction - Site Map

FUNCTIONS OF A COMPLEX VARIABLE
We may imagine them as lying on a circle of unit radius in the complex plane.
Special points on this circle are
ˆ 0
ˆ =2
ˆ
…1†
ˆ=2
…i†
…1†
…i†:
The reader should know these points at all times.
Sometimes it is easier to use the polar form in manipulations. For example, to
multiply two complex numbers, we multiply their moduli and add their phases; to
divide, we divide by the modulus and subtract the phase of the denominator:
zz 1 ˆ…re i †…r 1 e i 1
†ˆrr 1 e i…‡ 1† ;
z
ˆ rei
z 1 r 1 e i ˆ r e i…1† :
1 r 1
On the other hand to add two complex numbers we have to go back to the
Cartesian forms, add the components and revert to the polar form.
If we view a complex number z as a vector, then the multiplication of z by e i
(where is real) can be interpreted as a rotation of z counterclockwise through
angle ; and we can consider e i as an operator which acts on z to produce this
rotation. Similarly, the multiplication of two complex numbers represents a rotation
and a change of length: z 1 ˆ r 1 e i 1
; z 2 ˆ r 2 e i 2
, z 1 z 2 ˆ r 1 r 2 e i… 1‡ 2 † ; the new
complex number has length r 1 r 2 and phase 1 ‡ 2 .
Example 6.1
Find …1 ‡ i† 8 .
Solution: We
p
®rst write z in polar form: z ˆ 1 ‡ i ˆ r…cos ‡ i sin †, from which
we ®nd r ˆ
2 ;ˆ =4. Then
p
z ˆ
p
2 … cos =4 ‡ i sin =4†
ˆ 2 e i=4 :
Thus
p
…1 ‡ i† 8 ˆ… 2 e i=4 † 8 ˆ 16e 2i ˆ 16:
Example 6.2
Show that
p
1 ‡ !
3 i
p
10ˆ
1 1 p
3 i 2 ‡ i 3
2 :
236