In their paper http://journals.aps.org/prd/abstract/10.1103/PhysRevD.42.3500 (Classical and quantal nonrelativistic Chern-Simons theory) Jackiw and Pi introduced an unusual identity involving two-dimensional delta-function, "which may very well stretch mathematical convention": $$(\partial_x\partial_y-\partial_y\partial_x)\arctan{\frac{y}{x}}=2\pi\delta^{(2)}(x,y).$$ Some further justification of this identity they discuss in http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.66.2682
What is the proper mathematical framework for such kind of identities? Unlike the delta-function, it cannot be a distribution theory, because for distributions partial derivatives commute.