 Find where the M ̈obius transformation f (z) = z + 1
z − i maps the unit circle, the real axis, and
the imaginary axis. Also draw graphs of the unit circle and real and imaginary axes in the
z-plane and their images in the w-plane.
• Note that you only need to calculate the images of 0, −1, 1, −i, i, and ∞ (note that
f (∞) = limz→∞ f (z), assuming the limit exists).
• Remember that M ̈obius transformations preserve angles.


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.

Find where the Möbius transformation f(z) = (z + 1) / (z - i) maps the unit circle, the real axis, and the imaginary axis. Also draw graphs of the unit circle and real and imaginary axes in the z-plane and their images in the w-plane. Calculate the images of 0, -1, 1, -i, i, and ∞.

Explore how the Möbius transformation f(z) = (z + 1) / (z - i) maps the unit circle, real axis, and imaginary axis. Learn the mathematical concepts behind conformal mapping and calculate the images of key points such as 0, -1, 1, -i, i, and ∞. Suitable for undergraduate students studying complex analysis.

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