 Solution

For a holomorphic function, compute its complex derivative along the real and imaginary directions. The real direction gives $f^{\prime }=u_x+i{v}_x$, while division by an imaginary increment gives $f^{\prime }=v_y-i{u}_y$. Equating real and imaginary parts proves the Cauchy-Riemann equations:For the specified real part, $u_x=-e^{-x}\cos y$ and $u_y=-e^{-x}\sin y$. Integrating $v_y=u_x$ gives $v=-e^{-x}\sin y+C(x)$; the other relation in the Cauchy-Riemann equations then forces $C^{\prime }(x)=0$. Thus the harmonic conjugate and the holomorphic function are