  A complex notion of backward stochastic differential equation (BSDE) is
proposed in this paper to give a probabilistic interpretation for linear first
order complex partial differential equation (PDE). By the uniqueness and
existence of regular solutions to complex BSDE, we deduce that there exists a
unique classical solution $\{\mathbb{U}(t,x)$ to complex PDE and
$\{\mathbb{U}(t,x)$ is analytic in $x$ for each $t$. Thus we extend the well
known real Feynman-Kac formula to a complex version. It is stressed that our
complex BSDE corresponds to a linear PDE without the second order term.
