Adjoint source-to-far-field operator
= Adjoint source-to-far-field operator
{title2=$T_k^*$}
With complex <inner products> linear in the first argument, the <adjoint operator> of the <source-to-far-field operator at fixed frequency> is
$$
(T_k^*g)(\mathbf r)=-\frac1{4\pi}\int_{S_1}g(\boldsymbol\theta)e^{ik\boldsymbol\theta\cdot\mathbf r}\,dS(\boldsymbol\theta).
$$
It is a normalized <Herglotz wave function>. Complex conjugation reverses the phase sign, while retaining the real factor $-1/(4\pi)$.