If we let $f(z)=(1-e^{1/z})^{-1} ,$ where $z$ is complex, I'm trying to use the Cauchy-Riemann equations to determine if $f$ is holomorphic. So I need to separate it into real and imaginary functions of $x$ and $y$ ($u(x,y)$ , $v(x,y)$ respectively), then partially differentiate.
So far I have
$u(x,y)=\frac {1-e^{\frac {x}{x^{2}+y^{2}}} \cos(\frac{y}{x^{2}+y^{2}})}{1-2e^{x} \cos(\frac{y}{x^{2}+y^{2}})+e^{\frac {1}{x^{2}+y^{2}}}} $
$v(x,y)=\frac {-e^{\frac {x}{x^{2}+y^{2}}} \sin(\frac{y}{x^{2}+y^{2}})}{1-2e^{x} \cos(\frac{y}{x^{2}+y^{2}})+e^{\frac {1}{x^{2}+y^{2}}}} $
This looks a bit of a nightmare to differentiate 4 times for a small part of a question, can anyone see how a quicker way to use the C-R equations to determine if $f$ is holomorphic?