Draw a line in the complex plane. What is the image of that line when you apply the exponential function?

A line through *w* with direction *z* is the set of points *w* + *tz* where *w* and *z* are complex and *t* ranges over the real numbers. The image of this line is

exp(*w*+ *tz*) = exp(*w*) exp(*tz*)

and we can see from this that character of the image mostly depends on *z*. We can first plot exp(*tz*) then rotate and stretch or shrink the result by multiplying by exp(*w*).

If *z* is real, so the line is horizontal, then the image of the line is a straight line.

If *z* is purely imaginary, so the line is vertical, then the image of the line is a circle.

But for a general value of *z*, corresponding to neither a perfectly horizontal nor perfectly vertical line, the image is a spiral.

Since the imaginary part of z is not zero, the image curves. And since the real part of *z* is not zero, the image spirals into the origin, either as *t* goes to infinity or to negative infinity, depending on the sign of the real part of *z*.

Here’s an example, the image of the line through (3, 0) with slope 2.