the Mobius transformation f(z)=z-i/z+I  maps  the circle |z|=1 to a circle ........ true or false

The statement is **true**. 

The Möbius transformation \( f(z) = \frac{z - i}{z + i} \) maps the circle \( |z| = 1 \) (the unit circle) to another circle. Möbius transformations generally map circles and lines in the complex plane to other circles and lines, with the unit circle often being transformed into a circle (though in some cases it might become a line).

If you want to explore this further or confirm details with a more specific derivation, feel free to ask!

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Here are five related questions to expand on this:
1. How do you derive the image of the unit circle under a Möbius transformation?
2. What are the general properties of Möbius transformations?
3. How does a Möbius transformation map the entire complex plane?
4. Can a Möbius transformation map a line to a circle?
5. How do fixed points behave under a Möbius transformation?

**Tip:** Möbius transformations can always be written as \( f(z) = \frac{az + b}{cz + d} \), and their properties are greatly influenced by the coefficients \( a, b, c, d \).

The Mobius transformation f(z)=z-i/z+I maps the circle |z|=1 to a circle. True or false?

Discover how the Möbius transformation f(z) = (z - i) / (z + i) maps the unit circle |z|=1 to another circle in the complex plane. Learn about the properties of Möbius transformations, their impact on circles, and relevant theorems in complex analysis.

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