# Complex fuction - is it analytic

1. Nov 28, 2005

### leila

Ok, here is the question

determine whether f(z)=(1+z)/(1-z) is analytic or otherwise. Unfortunetly I am having problems with the maths. So far I have substituted z=x+iy and got

1+x-iy/1-x-iy and if i let a=x+1 and b=1-x then that simplifies my problem to

a+iy/b-iy

so now i have to rearrange this fraction so that i have a term with i and a term without.

For the life of me I can't figure out how to actually do that

Any help would be much appreciated

Leila

2. Nov 28, 2005

### HallsofIvy

Analytic where? f(z) is not even defined at z= 1 and so cannot be analytic there. It is a simple rational function and so is analytic everywhere except at z= 1.

Apparently you are trying to use the Cauchy-Riemann equations to show that. To convert a fraction such as (a+iy)/(b-iy) multiply both numerator and denominator by the complex conjugate of the denominator:
$$\frac{a+iy}{b-iy}= \frac{a+iy}{a-iy}\frac{a+iy}{a+iy}= \frac{a^2+ 2ayi- y^2}{a^2+ y^2}= \frac{a^2- y^2}{a^2+y^2}+ \frac{2ay}{a^2+ y^2}i$$.