Electric field of an accelerating charge?

The field of a charge accelerating in an arbitrary way is very difficult to derive. There is only one textbook that I know of which gives a full - well, almost full - derivation. This is Feynman's Lectures on Physics, Vol II, Ch.21. He didn't actually complete the derivation of his formula, as the time alloted for the lecture wasn't long enough, but indicated generally what needed to be done for completion. The formula is too complicated for me to reproduce here, and the proof is something you'd better look up yourself. The field of a charge in uniform motion is more tractable, but even this requires special relativity to derive. Good textbooks usually treat the case of a charge executing simple harmonic motion, in view of the many applications of the results, but mathematically rigorous proofs could be tricky, and subtle points of uniqueness are often glossed over.