1. The problem statement, all variables and given/known data Suppose z = x + iy. Where are the following functions diﬀerentiable? Where are they holomorphic? Which are entire? the function is f(z) = e-xe-iy 2. Relevant equations ∂u/∂x = ∂v/∂y ∂u/∂y = -∂v/∂x 3. The attempt at a solution f(z) = e-xe-iy I convert it to polar form: f(z) = cos(xy) + isin(xy) then i set u as the real part of the equation, and v as the complex part of the equation. u = cos(xy) v = isin(xy) ∂u/∂x = -ysin(xy) ∂v/∂y= ixcos(xy) then ∂u/∂x =/= ∂v/∂y so the function is not holomorphic. I think i did this wrong because i've done this for the majority of the problem set, and every function is coming up as not holomorphic. Is my work correct? EDIT: I did it again and here is my new work, my other work was wrong because i multiplied exponents wrong f(z) = e-xe-iy = (cos(-x) + isin(-x)) * (cos(-y) + isin(-y)) = cos(-x)cos(-y) + icos(-y)sin(-x) + (-1)sin(-x)(sin-y) + isin(-x)cos(-y) so Re(f(z)) = cos(-x)cos(-y) - sin(-x)sin(-y) = u Im(f(z)) = cos(-y)sin(-x) + cos(-x)sin(-y) = v then ux = -(-sin(-x)cos(-y)) + cos(-x)sin(-y) ux = sin(-x)cos(-y) + cos(-x)sin(-y) and vy = - (-sin(-y)sin(-x)) + cos(-x)cos(-y) vy = sin(-y)sin(-x) - cos(-x)cos(-y) so ux =/= vy so f(z) is not holomorphic, and is not differentiable at all points( but may be differentiable at some points ), and is not entire because it is not differentiable at all points.