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Unit 7: Transformations and Conformal Mappings

Notes

Notes The angle of rotation and the scalar factor (linear magnification) can change from
point to point. We note that they are 0 and 2 respectively, at the point z = 1, since f (1) = 2,
where the curves C and C are the same as above and the non-negative x-axis (C ) is
2
3
2
transformed into the non-negative u-axis (C ).
3
Example: Discuss the mapping w = z , where a is a positive real number.
a
Solution. Denoting z and w in polar as z = re , w = re , the mapping gives r = r ,  = a.
a
i
i
Thus the radii vectors are raised to the power a and the angles with vertices at the origin are
multiplied by the factor a. If a > 1, distinct lines through the origin in the zplane are not mapped
onto distinct lines through the origin in the w-plane, since, e.g. the straight line through the
origin at an angle 2 to the real axis of the z-plane is mapped onto a line through the origin in
a
dw
the wplane at an angle 2 to the real axis i.e. the positive real axis itself. Further, = az ,
a1
dz
which vanishes at the origin if a > 1 and has a singularity at the origin if a < 1. Hence, the
mapping is conformal and the angles are therefore preserved, excepting at the origin. Similarly
the mapping w = e is conformal.
z

Example: Prove that the quadrant | z | < 1, 0 < arg z < is mapped conformally onto
2
4
a domain in the w-plane by the transformation w = 2 .
(z 1)

4
Solution. If w = f(z) = 2 , then f (z) is finite and does not vanish in the given quadrant. Hence,
(z 1)

the mapping w = f(z) is conformal and the quadrant is mapped onto a domain in the w-plane
provided w does not assume any value twice i.e. distinct points of the quadrant are mapped to
4 4
distinct points of the w-plane. We show that this indeed is true. If possible, let 2 = 2 ,
(z  1) (z  1)
2
1
where z  z and both z and z belong to the quadrant in the z-plane. Then, since z  z , we have
1
1
2
2
1
2
(z  z ) (z + z + 2) = 0
2
1
1
2
 z + z + 2 = 0 i.e. z = z  2. But since z belongs to the quadrant, z  2 does not, which
1
2
1
2
2
2
contradicts the assumption that z belongs to the quadrant. Hence w does not assume any value
1
twice.
7.3 Summary
Here, we shall study how various curves and regions are mapped by elementary analytic

function. We shall work in  i.e. the extended complex plane. We start with the linear

function.
w = Az (1)
where A is non-zero complex constant and z  0. We write A and z in exponential form as
A = ae , z = re i
i
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