Please see the attached file for full problem description.
3. Use Cauchy-Riemann equations and the given theorem to show that the function
f (z) = e^z
is not analytic anywhere.
Theorem: Suppose that
f (z) = u (x, y) + i v (x, y)
and that f'(z) exists at a point z0 = x0 + i y0. Then the first-order partial derivatives of u and v must exist at (x0, y0), and they must satisfy the Cauchy-Riemann equations
ux = vy, uy = - vx
there. Also, f ΄(z0) can be written
f ΄(z0) = ux + i vx,
where these partial derivatives are to be evaluated at (x0, y0).
Cauchy-Riemann Equations and First-Order Partial Derivatives are investigated. The solution is detailed and well presented.