Solution:
\( \displaylines{\text{The Cauchy-Riemann equations state: }\\ u_{x}=v_{y}\text{ and }u_{y}=-v_{x}\\ \text{For }f\left(z\right)=e^{z}\text{ and }g\left(z\right)=e^{\bar{z}}\text{ we have}\\ f=e^{x}e^{iy}\text{ and }g=e^{x}e^{-iy}\\ f=e^{x}\cos y+ie^{x}\sin y\text{ and }g=e^{x}\cos y-ie^{x}\sin y\\ \text{For f:}\\ u_{x}=e^{x}(cosy)\\ u_{y}=e^{x}\left(-siny\right)\\ v_{x}=e^{x}(siny)\\ v_{y}=e^{x}(cosy)\\ \text{Which satisfy the Cauchy-Riemann equations.}\\ \\ \text{However, for g:}\\ u_{x}=e^{x}(cosy)\\ u_{y}=-e^{x}(siny)\\ v_{x}=-e^{x}(siny)\\ v_{y}=-e^{x}(cosy)\\ \text{where because of the negative sign, these do not satisfy the Cauchy-Riemann equations.}\\ \\ f\text{ is analytic, while }g\text{ is not.}} \)