u(x,y) and v(x,y) satisfy the Caunchy-Riemann equations

Given the function: f(z) = z/(z-1)

I have made it into the form:

u(x,y) + I * v(x,y)

Caunchy-Riemann equations are only satisfied when y=0 according to my calculations.

Is this correct? Shoudn't the C.R equations hold everywhere except x=1 ? as from the above f(z) there is a singularity at z=1

1 Attachment(s)

Re: u(x,y) and v(x,y) satisfy the Caunchy-Riemann equations

they appear to be be satisfied everywhere though there is a pole at x=1

Attachment 30379

Re: u(x,y) and v(x,y) satisfy the Caunchy-Riemann equations

Thanks for that.

I made a algebra mistake in my workings.