|
Complex
Analysis. FreeTutorial
11.1
Division as multiplication and reciprocation
11.2 The modulus and argument
of the quotient
11.2.1
Example of division
Top
11.1 Division as multiplication and reciprocation
Let us replace division
z1/z2
with multiplication z1(1/z2),
where 1/z2
is the reciprocal of z2.
Step1: Construct conjugate
 2
= (x2,
-y2).
Step2: Construct reciprocal
 .
Step3: Perform multiplication
z1(1/z2)
= z1/z2.
Such way the division can be compounded
from multiplication and reciprocation.
Figure 1.18 shows all steps.
Figure 1.18 Division
of the complex numbers z1/z2
Top
11.2 The modulus and argument of the quotient
Let us consider two complex numbers z1
and z2
in a polar form.
z1
= r1(cos +i
sin),
z2
= r2(cos +i
sin).
It was shown (1.18) that their product
z1z2
has the modulus r1r2
and the argument +.
We have replaced division z1/z2
with multiplication z1(1/z2).
Let 1/z2
= z3
.
z3
= r3(cos +i
sin).
Then
z1/z2
=
z1z3
=
r1r3[(cos(+)
+ i(sin(+)].
The modulus r3
is the modulus of
reciprocal 1/z2.
r3
= |1/z2|.
The argument
is the argument of reciprocal 1/z2.
=
arg(1/z2)
= - arg(z2).
The quotient z1/z2
has the modulus |z1|/|z2|
and the argument {Arg(z1)
- Arg(z2)}.
|z1/z2|
= |z1|/|z2|,
(1.21)
Arg(z1/z2)
= Arg(z1)
- Arg(z2).
(1.22)
Top
11.2.1
Example of division
z1
= 2 + i ,
z2=
2 + i .
Find modulus |z1/z2|
and arg(z1/z2).
Construct z1/z2
using multiplication and reciprocation.
|z1|
= 4,
|z2|
=  ,
|z1/z2|
= |z1|/|z2|
=  .
arg(z1)
= 60º,
arg(z2)
= 30º.
arg(z1/z2)
= arg(z1)
- arg(z2)
= 60º
- 30º= 30º
z1/z2
= (cos(30º)+
i sin(30º)).
Let us construct z1/z2
using multiplication and reciprocation.
Step1: Construct conjugate
2.
|2|
= ,
arg(2)
= -30º.
Step2: Calculate |z2|2
and construct reciprocal of z2.
|z2|2
= ()2
= 16/3,
Reciprocal
has polar coordinates ( ,
-30º).
Step3: Perform multiplication
z1(1/z2)
= z1/z2.
z1(1/z2)
= (4, 60º)(,
-30º)
= (,
30º).
z1/z2
= (cos(30º)+
i sin(30º)).
See Figure1.19.
Figure
1.19 Example z1/z2
by
Tetyana Butler
|
