An exponential map is a conformal map of the form f(z) = ez.
Let z = x + iy.
⇒ f(z) = ex + iy = exeiy
⇒ The norm of f(z) = ex, argument f(z) = y.
Horizontal lines in the cartesian grid have a constant y value ⇒ the argument of the map of the horizontal line is also constant while the norm changes with ex ⇒ horizontal lines are mapped to rays with argument y.
Similarily, vertical lines have a constant x component ⇒ the map of vetical lines have a constant norm ⇒ vertical lines get mapped to circles centered at the origin with radius ex.
A Little Note About Branch Cuts:
The inverse map of f(z) = ez is f-1 = log(z). In this applet log(z) is defined to be log|z| + i Arg z, where Arg z ∈ (-π, π]. This means that f-1(z) has a branch cut along the negative real axis. If you try to draw a line across the branch cut Arg z will jump from -π to π (or vice versa depending on the direction you cross the cut...we'll leave that for you to figure out). Thus log(z) is not continuous along the branch cut.
Please draw slowly across the branch cut in the exponential grid!
Use the exotic applet below to do the following exercises:
Draw horizontal lines in the domain. How do they look in the image?
Same but with vertical lines.
Draw a circle in the image. How does it look in the domain?
Same, but draw the circle in the domain.
Draw a ray through the origin in the domain. What shape does it make in the image?
Using the applet.