https://r2.erweima.ai/mathbot/image/1725502257566ut663nnh.jpg

The function shown in the image is:

\[
f_8(z) = \frac{z - 1}{z + 1}
\]

This is a Möbius transformation, a type of complex function. Möbius transformations map points on the complex plane and are known for preserving angles and transforming circles into other circles or straight lines. 

### Key Properties:
1. **Singularity/Undefined Point**: The function is undefined at \( z = -1 \) because the denominator becomes zero.
2. **Fixed Points**: Solving for \( z \) when \( f_8(z) = z \) can determine the fixed points of the function.
3. **Behavior at Infinity**: Evaluating \( f_8(z) \) as \( z \to \infty \), the function approaches 1.
4. **Symmetry**: This transformation has symmetry about certain axes in the complex plane.

Would you like to explore any of these points in more detail or have further questions?

Here are five possible questions for expanding on this information:
1. How do you calculate the fixed points of this Möbius transformation?
2. What is the geometric interpretation of Möbius transformations on the complex plane?
3. How does the function behave near \( z = -1 \), where it is undefined?
4. How can you apply this transformation to specific points, like \( z = i \)?
5. Can you generalize the form of Möbius transformations to other functions?

**Tip**: To analyze Möbius transformations, check how they map key points like 0, 1, and infinity.

What is the function shown in the image and what are its key properties?

Explore the Möbius transformation f(z) = (z - 1) / (z + 1), its key properties including singularity points, fixed points, and behavior at infinity. This page is suitable for undergraduate students studying complex analysis.

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