Complex Analysis - Maths KU

134 Chapter 2 Complex Functions and Mappings
6
5
4
3
2
1
y
–2 –1 1 2 3 4 5 6
–1
–2
Figure 2.59 Vectors in the vector field
f(z) =z 2
y
4
3
2
1
1
2
Figure 2.60 Mathematica plot of the
vector field f(z) =iy
y
4
3
2
1
1
2
Figure 2.61 Mathematica plot of the
normalized vector field f(z) =iy
3
3
4
4
x
x
x
When plotting a vector field F associated with a complex function f it
is helpful to note that plotting the vector F(x, y) with initial point (x, y) is
equivalent to plotting the vector representation of the complex number f(z)
with initial point z. We illustrate this remark in the following example.
EXAMPLE 1 Plotting Vectors in a Vector Field
Plot the vectors in the vector field f(z) =z 2 corresponding to the points
z =1, 2+i, 1+i, and i.
Solution Bya straightforward computation we find that:
f(1) = 1 2 =1, f(2 + i) = (2 + i) 2 =3+4i,
f(i) =i 2 = −1, and f(1 + i) = (1 + i) 2 =2i.
This implies that in the vector field f(z) =z 2 we have the vector representations
of the complex numbers 1, 3+4i, −1, and 2i plotted with initial points
at 1, 2 + i, i, and 1 + i, respectively. These vectors are shown in Figure 2.59.
Use of Computers Plotting vector fields byhand is a simple but
tedious procedure. Fortunately, computer algebra systems such as Mathematica
and Maple have built-in commands to plot two-dimensional vector
fields. In Figure 2.60, the vector field f(z) =iy has been plotted using the
PlotVectorField command in Mathematica. Observe that the lengths of the
vectors in the Mathematica plot are much smaller than theyshould be. For
example at, say, z =1+i we have f(1 + i) =i, but the vector plotted at
z =1+i does not have length 1. The reason for this is that Mathematica
scales the vectors in a vector field in order to create a nicer image (in particular,
Mathematica scales to ensure that no vectors overlap). Therefore, the
lengths of the vectors in Figure 2.60 do not accuratelyrepresents the absolute
lengths of vectors in this vector field. The vectors in Figure 2.60 do, however,
accuratelyrepresent the relative lengths of the vectors in the vector field. ‡
In manyapplications the primaryinterest is in the directions and not the
magnitudes of the vectors in a vector field. For example, in the forthcoming
discussion we will be concerned with determining the paths along which particles
move in a fluid flow. For this type of application, we can use a normalized
vector field. In a normalized vector field all vectors are scaled to have the
same length. Figure 2.61 displays a normalized vector field for f(z) =iy created
using the ScaleFunction option with the PlotVectorField command
in Mathematica. Compare with Figure 2.60.
‡ For more information on plotting vector fields in Mathematica, refer to the technical
report Guide to Standard Mathematical Packages published by Wolfram Research.