The graphs above show the transformations of the unit circle, real axis, and imaginary axis under the Möbius transformation $f(z) = \frac{z + 1}{z - i}$.

• z-plane (left):

  • The unit circle is plotted in blue.
  • The real axis is plotted in red.
  • The imaginary axis is plotted in green.

• w-plane (right):

  • The transformed image of the unit circle is plotted in blue.
  • The transformed image of the real axis is in red.
  • The transformed image of the imaginary axis is in green.

The Möbius transformation preserves angles, and the unit circle has been mapped into a line-like structure. The real axis and imaginary axis have been transformed into curves in the $w$-plane, likely circular arcs.

Would you like to dive into any of these transformations in more detail, or do you have any questions?

5 Related Questions:

1. What happens to other points inside the unit circle under this Möbius transformation?
2. How would the transformation behave if $z$ approaches 0 from different directions?
3. Can you calculate the inverse of this Möbius transformation?
4. What is the effect of Möbius transformations on the upper half-plane?
5. How are Möbius transformations used in complex analysis and geometry?

Tip:

Möbius transformations are conformal, meaning they preserve angles at intersections of curves, making them powerful tools in geometric function theory.