= Solution
For a <holomorphic function>, compute its complex <derivative> along the real and imaginary directions. The real direction gives $f'=u_x+iv_x$, while division by an imaginary increment gives $f'=v_y-iu_y$. Equating real and imaginary parts proves the <Cauchy-Riemann equations>:
$$
\boxed{u_x=v_y,\qquad u_y=-v_x.}
$$
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 $C'(x)=0$. Thus the <harmonic conjugate> and the <holomorphic function> are
$$
\boxed{v(x,y)=-e^{-x}\sin y+C,\qquad f(z)=e^{-z}+iC,\quad C\in\mathbb R.}
$$
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