$\displaystyle Since,\ f(z)=\ u+iv\ be\ analytic\ \\[2ex] \displaystyle By\ Cauchy\ rehmans\ equations \\[2ex] \displaystyle u_x=v_y\ and\ u_y=-v_x\ \ \ ……(A) \\[2ex] \displaystyle Differentiating\ u-v\ =\ e^x\ \ \left(cos\ y\ -\ sin\ y\right)\\[2ex]partially\ w.r.t\ x \\[2ex] \displaystyle u_x-v_x=e^x(\cos{y-\sin y} \ldots{} \ldots{} (B)\\[3ex] \displaystyle Differentiating\ u-v\ =\ e^x\ \ \left(\cos\ y\ -\sin\ …