arXiv:1308.2262v1 [math-ph] 10 Aug 2013 COMMENT ON ‘COMMENT ON ‘SOME NOVEL DELTA-FUNCTION IDENTITIES’ BY CHARLES P FRAHM (AM. J. PHYS 51 826–9 (1983))’ BY J. FRANKLIN (AM. J. PHYS 78 1225–26 (2010)) YUNYUN YANG AND RICARDO ESTRADA Abstract. We prove the formula ∂ *2 (r -1 ) ∂x i ∂x j = ( 3x i x j − δ ij r 2 ) P f ( r -5 ) +4π (δ ij − 4n i n j ) δ * , for the second order thick derivatives of r -1 in R 3 , where δ * is a thick delta of order 0. This formula generalizes the well known Frahm formulas for the distributional derivatives of r -1 . Our proof shows that the extended formula given in “Comment on ‘Some novel delta-function identities’ by Charles P Frahm (Am. J. Phys 51 826–9 (1983)),” Am. J. Phys 78 1225–26 (2010), is not correct. 1. Introduction The derivatives of 1/r have been given great consideration, because the inverse-square field is very important in classic field theories when considering a point source. The potential of an inverse-square field usually involves an 1/r term. And the field equations typically involves a derivative term with respect to the field. For example, (1.1) ∂ i E i =4πρ(r) , where E i corresponds to the electric field provided by a point charge q, (1.2) E i = qn i /r 2 . Since 1/r has a singular point at the origin – r is the radial coordinate – we cannot treat its derivatives there as usual derivatives, but as distributional derivatives 1 . The second order distributional derivatives of 1/r were given by Frahm [3], (1.3) ∂ 2 ∂x i ∂x j  1 r  = 3x i x j − δ ij r 2 r 5 −  4π 3  δ ij δ (r) , where one can find a derivation; other proofs can be found in the textbooks [6], [7]. Formulas (1.3) are correct distributional identities, yet they have been questioned by several authors 2 . One of them was Franklin [4], who proposed to “correct” the familiar 1991 Mathematics Subject Classification. 46F10. Key words and phrases. Thick delta functions, thick distributions, Frahm formulas. The authors gratefully acknowledge support from NSF, through grant number 0968448. 1 We have followed the convention introduced by the late Professor Farassat [2] of denoting distribu- tional derivatives with an overbar, to distinguish them from the ordinary derivatives. 2 The distribution (3x i x j − r 2 δ ij )r -5 is a little complicated since the integral  R 3 (3x i x j − r 2 δ ij )r -5 φ (x)dx is divergent, in general. Its definition is  3x i x j − r 2 δ ij r 5 ,φ (x)  = lim ε→0 +  |x|≥ε  3x i x j − r 2 δ ij r 5  φ 1