# Homework Help: Properties of the modulus - complex variables question

1. Sep 21, 2009

### jaejoon89

Given
f(z) = (z+1) / (z-1) for z not equal to 1

My teacher wrote
|f(z)| = |x+1+iy| / |x-1+iy| = sqrt((x+1)^2 +1) / sqrt((x-1)^2 + 1)

How do the values within the modulus work out to the right hand side? I can't figure it out.

2. Sep 21, 2009

### snipez90

That's the definition of the modulus. You square the real part and then you square the imaginary part and add the two together and finally you take the square root. But I think there is an error since the imaginary part of z = x + iy is y, not 1.

3. Sep 21, 2009

### futurebird

$$f(z) = \frac{z+1}{z-1}$$

$$|f(x+iy)| = \frac{|x+iy+1|}{|x+iy-1|}$$

$$= \frac{|x+1+iy|}{|x-1+iy|}$$
$$= \frac{\sqrt{(x+1)^2+(iy)^2}}{\sqrt{(x-1)^2+(iy)^2}}$$
$$= \frac{\sqrt{(x^2+2x+1)+(-y^2)}}{\sqrt{(x^2-2x+1)+(-y^2)}}$$
$$= \frac{\sqrt{(x^2+2x+1)+(-y^2)}}{\sqrt{(x^2-2x+1)+(-y^2)}}$$

?