  For a strongly elliptic pseudodifferential operator $L$ of order $2a$
($0<a<1$) with real kernel, we show an integration-by-parts formula for
solutions of the homogeneous Dirichlet problem, in the model case where the
operator is $x$-independent with homogeneous symbol, considered on the
halfspace $R^n_+$. The new aspect compared to $(-\Delta )^a$ is that $L$ is
nonsymmetric, having both an even and an odd part. Hence it satisfies a $\mu
$-transmission condition where generally $\mu \ne a$.
  We present a complex method, relying on a factorization in factors
holomorphic in $\xi_n$ in the lower or upper complex halfplane, using
order-reducing operators combined with a decomposition principle originating
from Wiener and Hopf. This is in contrast to a real, computational method
presented very recently by Dipierro, Ros-Oton, Serra and Valdinoci. Our method
allows $\mu $ in a larger range than they consider.
  Another new contribution is the (model) study of "large" solutions of
nonhomogeneous Dirichlet problems when $\mu >0$. Here we deduce a "halfways
Green's formula" for $L$: $$ \int_{R^n_+} Lu\,\bar
v\,dx-\int_{R^n_+}u\,\overline{ L^*v}\,dx=c\int_{R^{n-1}}\gamma_0(u/x_n^{\mu -1
})\,{\gamma_0(\bar v/x_n^{\mu ^*})}\, dx', $$ when $u$ solves a nonhomogeneous
Dirichlet problem for $L$, and $v$ solves a homogeneous Dirichlet problem for
$L^*$; $\mu ^*=2a-\mu $. Finally, we show a full Green's formula, when both $u$
and $v$ solve nonhomogeneous Dirichlet problems; here both Dirichlet and
Neumann traces of $u$ and $v$ enter, as well as a first-order
pseudodifferential operator over the boundary.
