Polynomial source function moments for emergent intensity ($A_j=j!a_j$)

For a semi-infinite plane-parallel atmosphere with polynomial source function $S(\tau )=\sum a_j{\tau }^j$, the outward specific intensity is ${\int }_0^{\infty }S(\tau )e^{-\tau /\mu }d\tau /\mu$. Substitution $u=\tau /\mu$ gives coefficients $A_j=j!a_j$ in the surface polynomial in $\mu$. The deep boundary term must vanish. A local truncated Taylor series instead gives this contribution plus a remainder, so its surface polynomial is not automatically exact.