Complex
Analysis. FreeTutorial
10.1
Definition and notation of Reciprocals
10.1.1
Modulus and argument of Reciprocals
10.1.2
Geometric interpretation of Reciprocals
10.1 Definition and notation of Reciprocals
Any nonzero complex number
z
= x
+
yi
has a unique multiplicative inverse or
reciprocal 1/z
such that
z(1/z)
= 1.
1/z
= .
(1.16)
The reciprocal of zero is
undefined. The reciprocal of the complex
number z
is equal to its conjugate
,
divided by the square of the modulus of
the complex numbers z.
.
(1.17)
Example
17:
If z
=
4
+ 3i,
then what is the
reciprocal of z?
=
4
 3i;
z
= =
=
5;
z^{}^{2}
= 25;
= (4
 3i)/25
=
4/25

(3/25)i.
Example
18:
If z
= i,
then what is the
reciprocal of i?
= – i;
1/i
= =
((–i)/1)
= – i.
The reciprocal of i
is its own conjugate – i.
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10.1.1
Modulus and argument of Reciprocals
We remember that the conjugate
numbers have the same modulus and opposite
arguments.
z
= ,
arg(z)
=  arg().
The modulus of reciprocal of z
is equal to 1/z.
1/z
= 1/z.
The argument of reciprocal is equal to
argument of the conjugate number .
arg(1/z)
= arg()
=  arg(z).
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10.1.2
Geometric interpretation of reciprocals
The complex number z
= x
+
yi
with polar coordinates (r,)
has length r
> 0 and angle
.
Its reciprocal 1/z
with polar coordinates (1/r,
 )
has length 1/r
> 0
and angle  .
If r
> 1, then the
length of the reciprocal is 1/r
< 1.
If 0 <
r
< 1, then 1/r
< 1.
Figure 1.7 shows the reciprocal
1/z
of the complex number z.
Figure1.7
The reciprocal 1/z
The reciprocal 1/z
of the complex number z
can be visualized as its conjugate ,
devided by the square of the modulus of
the complex numbers
z.
by
Tetyana Butler
