Vector Potential

For T.E. radiation the components of the vector potential relative to the orthonormal basis are

$$(\hat A_\xi,\hat A_\tau,\hat A_r,\hat A_\theta ) = \left( 0,0,\frac{-1}{r} \frac{\partial\psi_F}{\partial \theta},\frac{\partial\psi_F}{\partial r} \right)$$

$$= \left( 0,0,~~~~0~~~~,\frac{\pm4\pi a^2r}{(2\xi\xi')^2} \left( \frac{u}{(u^2+1)^{3/2}} \dot q \mp \frac{1}{u^2+1} \ddot q \right) \right)$$ (79)

where an over-dot refers to the partial derivative

$$\dot q=\frac{\partial q (\tau\pm \sinh^{-1} u)}{\partial \tau} ~.$$