AdS scalar-field boundary asymptotics

A scalar field of mass $m$ in ${\mathrm{AdS}}_{d+1}$ behaves near the conformal boundary as $\varphi =z^{d-\Delta }J+z^{\Delta }A+\cdots$, where $m^2=\Delta (\Delta -d)$. The leading coefficient is the source and the normalizable coefficient determines the operator expectation value.