Straight lines under e^z

This image is an attempt to reproduce Figure [19], p. 81 of Needham by actually applying the function e^z to lines defined as arrays of points in the z plane. The general scheme of the image in the w plane is the same, but the image in Needham appears to be artificially contracted where Re[z] > 0 and artificially expanded where Re[z] < 0.

The more Re[z] decreases, the smaller the radius of the circles deriving from the vertical lines. We can see why by letting z = x + y i. Then w(z) = e^{x + y i} = e^xe^{y i}. We could write w(z) = re^{y i}, where r = e^x. If x is some large negative number, say -1000, we have r = 1/e¹⁰⁰⁰, showing that r is becoming small. Even at x = -4, we have r = 1/e⁴ ≡ 0.01831. At x = 0, r = 1, showing that when Re[z] < 0, w(z) lies within the unit circle. Consequently, the gray circles, corresponding to the gray vertical lines to the left of the y axis, are squeezed into the unit disk where they are almost invisible. Only four of the blue circles (Re[z] > 0) fit within the boundaries of this plot.

We conclude that positive movements of z on the x axis expand the distances between points in w by increasing |w| without changing Arg(w). Positive movements on the y axis wrap more points around the origin without changing the distances between successive points at a particular distance |w|. When we vary both x and y, we see a spiral.

The bold T is distorted much more than one would expect from Needham's figure. Consequently, to see that T preserves its relation to the small squares in their images in the w plane, we had to move it closer to the origin in the z plane and keep the winding of w within 2 π.