Determine all values of $z$ (if any) where $f(z) = e^{|z|}$ is analytic?

Question

So, I proceeded with the Cauchy-Riemann equations after setting $z = x+ iy$ and so $f(z) = e^{\sqrt{x^2 + y^2}}$, then I got the Cauchy-Riemann equations, but how can I proceed after this?

Answer 1

If a function is real valued and analytic on some open connected set, then it must be constant. This follows immediately from the Cauchy Riemann equations.

The function $f$ is non-constant and real valued for all $z$, hence it cannot be analytic.

Answer 2

If an analytic function is nonconstant on some open connected set, then it must be open, hence its image cannot be contained in $\mathbb{R}$.

Answer 3

Hint

$$u(x,y)=e^{\sqrt{x^2+y^2}} \,;\, v(x,y)=0$$

The CR equations tell you that if the function is differentiable you must have

$$u_x =0 ; u_y =0 \,.$$

Solve those two equations, and at all other points the function is NOT differentiable. You then must see what happens at the very few points (if any) where these equations hold.

Answer 4

I will interpret the question as asking for the set of points at which the function is complex differentiable, since the question about its analyticity on open subsets was already answered. The function $f$ is real valued, so it's derivative is also real valued. A real valued linear function is complex linear iff it is zero. Now, by using the chain rule we see that the (real) gradient of the function $f$ is nonzero at every point where the gradient is defined, that is, except at the origin. Hence, $f$ is not complex differentiable at any point.