= Exciting field of a conductivity inclusion
{title2=$E_{\mathrm{exc}}=E+e_{\mathrm{regular}}(y)$}
The exciting field excludes an inclusion's own singular dipole field. Other inclusions and periodic images supply the regular correction at its centre $y$. The angular mean of a self-dipole gradient on a concentric sphere vanishes, while the mean-value property recovers the regular field at the centre. Consequently a spherical average can recover the exciting field, even though the total pointwise field is not constant on that sphere.
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