Calculate the real and imaginary parts of the holomorphic function $f(z)=z^2\cos(z)-e^{z^3-z}$ and verify directly that each of these functions is harmonic.
I believe I know how to the question, however I'm finding it quite tedious and I'm not confident in my work. I have tried substituting $z=x+iy$ and separating the components into real and imaginary parts, but I'm left with an ugly equation (which I'm not sure how to separate). Is there any easier way? Thanks, any help is appreciated.