OK, so let's write $z = x + iy$, $F = u + iv$.

$F(x+iy) = e^{(x+iy)} + 2(x+iy) = e^x e^{iy} + 2x + 2iy$

Remember Euler's identity, $e^{iy} = \cos(y) + i\sin(y)$.

Multiply out, then combine all the real terms (u) and all the imaginary terms (v).

The second and third functions are comparatively easier. Just remember to bring all of the $i$ terms to the numerator when you simplify.