1550251515-Classical_Complex_Analysis__Gonzalez_

218 Chapter5

5.4 The Reciprocal Function w = 1/ z

This function is defined in C-{O}. Between moduli and arguments of the
variables we have the relations

lwllzl = 1 and argw = -argz
Hence the mapping defined by this function may be decomposed into an
inversion with respect to the unit circle followed by a symmetry with respect
to the real axis (or vice versa). That is, to obtain the point w corresponding
to a given z = re;e we may first construct its inverse z' = (1/r)e;e and
then its conjugate w = z' = (l/r)e-i^8 , or we may reverse steps (Fig. 5.4).

For the geometric construction of the inverse, see Section 1.10 part( d). It

should be noted that a pure inversion with respect to the unit circle is
defined by z' = l/z.
From geometry, we know that the inversion transformation preserves
the magnitude of the angles but reverses their orientation, and that the
same property has the symmetry. Hence it follows that under the mapping

defined by w = 1/ z, angles are preserved in magnitude and orientation.

Also, from the known properties of the inversion and the symmetry, the
following geometric propertied of the mapping w = 1/ z are easily obtained:

1. Straight lines through the origin are mapped into straight lines through
   the origin.


2. Circles not containing the origin are mapped into circles not containing
   the origin.


3. Straight lines not containing the origin are mapped into circles passing
   through the origin.

y

z

x

Fig. 5.4