An exponential map is a conformal map of the form f(z) = e

Let z = x + iy.

⇒ f(z) = e

⇒ The norm of f(z) = e

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 e

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 e

The inverse map of f(z) = e

Please draw slowly across the branch cut in the exponential grid!

- Ex.1

Draw horizontal lines in the domain. How do they look in the image?

Same but with vertical lines.

- Ex.2

Draw a circle in the image. How does it look in the domain?

Same, but draw the circle in the domain.

- Ex.2

Draw a ray through the origin in the domain. What shape does it make in the image?

Using the applet.

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