$f(z)=e^{y}e^{ix}$
$=e^{y}(\cos x+i\sin x)$
$u(x,y)=e^{y} \cos x$
$v(x,y)=e^{y}\sin x$
$\frac{\partial u}{\partial x}=-e^{y} \sin x$
$\frac{\partial v}{\partial x}=e^{y} \cos x$
$\frac{\partial u}{\partial y}=e^{y}\cos x$
$\frac{\partial v}{\partial y}=e^{y}\sin x$
We see that the Cauchy-Riemann equations are not satisfied anywhere in the z-plane except at $z=n\pi$ where n is an integer.
We need to check if $f^{'}(z)$ exists at $z=n\pi$
$f^{'}(z)=Lt _{\delta z -> 0}\frac{f(z+\delta z)-f(z)}{\delta z}$
$\delta z = \delta x + i \delta y$
Best Answer
Hint: $z=x+iy$ and
$$f(z)=e^{i\bar{z}}.$$
The C.R. equations are equivalent to the single equation...