Complex Analysis - Maths KU

y
2
1.5
1
0.5
–2 –1.5 –1 –0.5
–0.5
0.5 1 1.5
x
2
–1
–1.5
–2
(a) A triangle in the z-plane
v
2
1.5
1
0.5
S′
–2 –1.5 –1 –0.5
–0.5
0.5 1 1.5
u
2
–1
–1.5
–2
w = z 2
(b) The image of the triangle in (a)
Figure 2.22 The mapping w = z 2
S
2.4 Special Power Functions 85
Thus, we have shown that the image of triangle S shown in color in Figure
2.22(a) is the figure S ′ shown in black in Figure 2.22(b).
The Function z n ,n > 2 An analysis similar to that used for the
mapping w = z 2 can be applied to the mapping w = z n , n>2. Byreplacing
the symbol z with re iθ we obtain:
w = z n = r n e inθ . (5)
Consequently, if z and w = z n are plotted in the same copyof the complex
plane, then this mapping can be visualized as the process of magnifying or
contracting the modulus r of z to the modulus r n of w, and byrotating z
about the origin to increase an argument θ of z to an argument nθ of w.
We can use this description of w = z n to show that a rayemanating from
the origin and making an angle of φ radians with the positive x-axis is mapped
onto a rayemanating from the origin and making an angle of nφ radians with
the positive u-axis. This propertyis illustrated for the mapping w = z 3 in
Figure 2.23. Each rayshown in color in Figure 2.23(a) is mapped onto a ray
shown in black in Figure 2.23(b). Since the mapping w = z 3 increases the
argument of a point bya factor of 3, the raynearest the x-axis in the first
quadrant in Figure 2.23(a) is mapped onto the rayin the first quadrant in
Figure 2.23(b), and the remaining rayin the first quadrant in Figure 2.23(a) is
mapped onto the rayin the second quadrant in Figure 2.23(b). Similarly, the
raynearest the x-axis in the fourth quadrant in Figure 2.23(a) is mapped onto
the rayin the fourth quadrant in Figure 2.23(b), and the remaining rayin the
fourth quadrant of Figure 2.23(a) is mapped onto the rayin the third quadrant
in Figure 2.23(b).
2
1.5
1
0.5
–2 –1.5 –1 –0.5
–0.5
0.5 1 1.5 2
–1
–1.5
–2
y
x
(a) Rays in the z-plane (b) Images of the rays in (a)
Figure 2.23 The mapping w = z 3
w = z 3
v
2
1.5
1
0.5
–2 –1.5 –1 –0.5
–0.5
0.5 1 1.5 2
–1
–1.5
–2
u