  We prove, for a wide class of semilinear elliptic differential and
pseudodifferential equations in $\R^d$, that the solutions which are
sufficiently regular and have a certain decay at infinity extend to holomorphic
functions in sectors of $\mathbb{C}^d$, improving earlier results where the
extension was shown for a strip. Moreover, exponential decay for such extended
solutions is also proved. The results apply, in particular, to solitary wave
solutions of many classical nonlinear evolution equations.
