Multipole Formulae for Gravitational Lensing Shear and Flexion

Understand

The gravitational lensing equations for convergence, potential, shear, and flexion are simple in polar coordinates and separate under a multipole expansion once the shear and flexion spinors are rotated into a ``tangential'' basis.

• We use this to investigate whether the useful monopole aperture-mass shear formulae generalize to all multipoles and to flexions.
• We re-derive the result of Schneider and Bartelmann that the shear multipole m at radius R is completely determined by the mass multipole at R, plus specific moments Q^m_in and Q^m_out of the mass multipoles internal and external, respectively, to R.
• The m>=0 multipoles are independent of Q_out.