Page 25 - Calculus with Complex Numbers

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Ffgure 2.J
which gives on eliminating y,

I v?I= IRe w - 2.x21

which is the equation of a parabola with focus w = 0, directrix Re w = 2.x2 . This
parabola is the image of the line .x = constant.
Similarly, eliminating .x, we get

2 + ,2 = (u + 2,2)2

IwI- Iltew + 2y2I,

which is a parabola, againwith focus w = 0, but now with directrix Re w = -2y2 .
This is the image of the grid line y = constant.
Another example wllichreaders might liketo work out forthemselves is w = 1/z
whichtransforms the grid lines .x = constant y = constant in the z-planeto circles
through the origin with centres on the real and imaginal'y axes in the w-plane (see
Figure 2.3).

2.4 T he expone ntial function

For real variables the function y = ex has the graph illustrated in Figure 2.4.
For complex variables we have

x i
z
W = e = d x + iy = e e y