Derivatives of the Dirac Delta

20 Sep 2016

I’ve been thinking lately about derivatives of the Dirac delta function. In particular, how to obtain formulas like

$$\delta'(f(x)) = \text{sign}(f') \left[ \frac{1}{f^{'2}} \delta'(x-x_i) + \frac{f''}{f^{'3}} \delta(x-x_i) \right]$$

and similar, more complicated expressions for higher derivatives. In the above formula, the function $f(x)$ is assumed to have a single zero at $x = x_i$, and all of the derivatives on the right-hand side are evaluated at $x_i$. If $f(x)$ 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

$$\int_{-\infty}^{\infty} \delta''(\cos{x})\; e^{-x^2} \text{d}x .$$

Welcome!

11 Aug 2016

Hi everyone!

This is my new website. I am currently working on transferring things over from my old site on Wordpress. Be sure to check back over time as I accumulate more material here.