Easy way to check that real part of $e^{-\frac{1}{z^{4}}}$ is harmonic.

Question

Let $z=x+i y$ and $$ u(x,y) = Re \left(e^{-\frac{1}{z^{4}}}\right) ,~ \text{for}~ (x,y) \ne (0,0)$$ and $0$ otherwise. Then is there any short way to check that $u$ satisfies Laplace equation ?

I can apply brute force to check that given function is harmonic in $\mathbb{R}^{2}$, but I think there may be some easy way to check it is harmonic.

Answer 1

That function is holomorphic and the real part of a holomorphic functions is always harmonic.

Answer 2

function is not holomorphic at $z=0$.

$z=0$ is a point of essential singularity.