Question

Let f(z) be the Möbius transformation f(z)=i-z/i+z

Describe completely the image of the set z and draw a sketch.

Answer 1

The image of the set z: under the Möbius transformation f(z) = rac{i-z}{i+z} is {f(r) = rac{i(-1+r)}{r+i} : r is any real number}

Step-by-step explanation:

The Möbius transformation f(z) = rac{i-z}{i+z} can be used to describe the image of the set z.

When we substitute z = r, where r represents any real number, into the equation for f(z), we get f(r) = rac{i-r}{i+r}. By simplifying this expression, we find that f(r) = rac{i(-1+r)}{r+i}.

Therefore, the image of the set z is the set of complex numbers {f(r) = rac{i(-1+r)}{r+i} : r is any real number}.