Exploring Complex Mapping: Images of i, 1-i, and the Real and Imaginary Axes

[Stephen88]

Homework Statement

Let the Complex mapping
z → f(z) =(1 + z)/(1 − z)
1.What are the images of i and 1 − i and 2.What are the images of the real and the imaginary axes?

The Attempt at a Solution

For i we have f(i)=(1+i)/(1-i) since i(depending on the power) can be i,-i,1,-1=>0, (1+i)/(1-i),(1-i)/(1+i)
For 1 − i we have 1,-3,(2-i)/i,(2+i)/i.
Not sure about the second part..

 

[vela]

Homework Statement

Let the Complex mapping
z → f(z) =(1 + z)/(1 − z)
1.What are the images of i and 1 − i and 2.What are the images of the real and the imaginary axes?

The Attempt at a Solution

For i we have f(i)=(1+i)/(1-i) since i(depending on the power) can be i,-i,1,-1=>0, (1+i)/(1-i),(1-i)/(1+i)

I'm not sure why you're looking at the various powers of i. It only appears to the first power in your expression. Use the following method to simplify it:
$$f(i) = \frac{1+i}{1-i} = \frac{1+i}{1-i}\cdot\frac{1+i}{1+i} = \ ?$$

 

[Stephen88]

Sorry I was tired,thanks for the reply..how should I think about the both parts of the problem.?..Also.I"m getting -1 for f(i)

 

[vela]

That's still wrong. You should find f(i)=i.

If z=x+iy is a point on the imaginary axis, you know that x=0, so you want to find f(z)=f(iy). Can you take it from there?