For a boundary-independent source, the <AdS scalar-field boundary asymptotics> reduce to
$$
\phi(z)=J+z^5A+o(z^5).
$$
One may therefore extract the source and response directly:
$$
\boxed{J=\lim_{z\to0}\phi(z)},
\qquad
\boxed{A=\lim_{z\to0}z^{-5}\bigl(\phi(z)-J\bigr)
=\frac15\lim_{z\to0}z^{-4}\partial_z\phi}.
$$
The operator expectation value is proportional to $A$; choosing the boundary-operator normalization $\mathcal O=A$ gives the requested expressions. A conventional action normalization can instead multiply this relation by the fixed factor $2\Delta-d=5$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 354 / 1 / b / ii / Solution

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


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