Derivatives of the Dirac Delta
20 Sep 2016I’ve been thinking lately about derivatives of the Dirac delta function. In particular, how to obtain formulas like
and similar, more complicated expressions for higher derivatives. In the above formula, the function is assumed to have a single zero at , and all of the derivatives on the right-hand side are evaluated at . If would have multiple zeroes, the right-hand side would include a sum over all the zeroes.
Deriving these formulas isn’t that hard, but I’ve been trying to come up with some nice applications; perhaps they might be useful in the electrodynamics of infinite curves, rather than straight lines of charge. You can also use them to evaluate integrals such as