New representations of π and Dirac delta using the nonextensive-statistical-mechanics q-exponential function

Abstract

We present a generalization of the representation in plane waves of Dirac delta, δ(x)=(1/2π)∫−∞∞e−ikxdk, namely, δ(x)=[(2−q)/2π]∫−∞∞eq−ikxdk, using the non-extensive-statistical-mechanics q-exponential function, eqix≡[1+(1−q)ix]1/(1−q) with e1ix≡eix, x being any real number, for real values of q within the interval [1,2[. Concomitantly, with the development of these new representations of Dirac delta, we also present two new families of representations of the transcendental number π. Incidentally, we remark that the q-plane wave form which emerges, namely, eqikx, is normalizable for 1<q<3, in contrast to the standard one, eikx, which is not.