I have the following question:
Let $s(y)$ and $t(y)$ be real differentiable functions on $y$ with $-\infty < y < \infty$ satisfying $s(0) = 1$ and $t(0) = 0$, with the property that the complex function
$$f(x + iy) = e^x(s(y) + it(y))$$
is entire. Find $s(y)$ and $t(y)$ (with proof).
Proof: Notice that on the real line $f \equiv e^x$. By the identity Theorem, $f(x+iy) = e^{x+iy}$. Therefore, $s(y) = e^{iy}$ and $t(y) = 0$.
Does the above thinking make sense? Am I applying the identity Theorem properly? I originally tried using Cauchy Riemann Equations, but could not conclude anything. Any thoughts or suggestions would be greatly appreciated.