Page 81 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes
Example: Discuss the mapping w = z .
Solution. We observe that the given mapping replaces every point by its reflection in the real
axis. Hence, angles are conserved but their signs are changed and thus, the mapping is isogonal
but not conformal. If the mapping w = z is followed by a conformal transformation, then
resulting transformation of the form w = f( z ) is also isogonal but not conformal, where f(z) is
analytic function of z.

Example: Discuss the nature of the mapping w = z at the point z = 1 + i and examine its
2
effect on the lines Im z = Re z and Re z = 1 passing through that point.
Solution. We note that the argument of the derivative of f(z) = z at z = 1 + i is
2
[arg 2z] z = 1+ i = arg(2 + 2i) = /4
Hence, the tangent to each curve through z = 1 + i will be turned by the angle /4. The co-efficient
of linear magnification is |f (z)| at z = 1 + i, i.e. |2 + 2i| = 2 2 . The mapping is

w = z = x  y + 2ixy = u(x, y) + iv(x, y)
2
2
2
We observe that mapping is conformal at the point z = 1 + i, where the half lines y = x(y  0) and
x = 1(y  0) intersect. We denote these half lines by C and C , with positive sense upwards and
2
1
observe that the angle from C to C is /4 at their point of intersection. We have
2
1
u = x  y , v = 2xy
2
2
The half line C is transformed into the curve C given by
1
1
u = 0, v = 2y (y  0)
2
Thus, C is the upper half v  0 of the vaxis.
1
The half line C is transformed into the curve C represented by
2 2
u = 1  y , v = 2y (y  0)
2
Hence, C is the upper half of the parabola v = 4(u  1). We note that, in each case, the positive
2
2
sense of the image curve is upward.
For the image curve C ,
2
dv dv/dy 2 2
du  du/dy   2y   v
dv
In particular, = 1 when v = 2. Consequently, the angle from the image curve C to the
du 1
image curve C at the point w = f(1 + i) = 2i is  , as required by the conformality of the mapping
4
2
there.
'
y v C
1
C 2
C 1
/4
' 2i
/4 C
2
1+i
/2 C 3 /2 '
C
3
O 1 x O 1 u
74 LOVELY PROFESSIONAL UNIVERSITY