Trying to show any f(z)=u(x,y) + i*v(x,y) with the 3 properties:
1. f(x+0*i)=e^x,
2. f(z) is entire,
3. f ' (z)=f(z) for all z,
has to be f(z)=(e^x) * (cos y + i*siny).
The hint in the book (Complex Variables and Applications, 6 ed., Churchill+Brown) says:
(a) Get u=u sub x and v=v sub...