New representations of pi 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, $\delta (x)=(1/2\pi ){\int }_{-\infty }^{\infty }{e}^{-ikx}dk$, namely $\delta (x)=(2-q)/(2\pi ){\int }_{-\infty }^{\infty }{e}_q^{-ikx}dk$, using the nonextensive-statistical-mechanics $q$-exponential function, $e_q^{ix}\equiv [1+(1-q)ix{]}^{1/(1-q)}$ with $e_1^{ix}\equiv e^{ix}$, being $x$ 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 $\pi$. Incidentally, we remark that the $q$-plane wave form which emerges, namely $e_q^{ikx}$, is normalizable for $1<q<3$, in contrast with the standard one, $e^{ikx}$, which is not.

13 pages, 6 figures. Accepted for publication in the Journal of Mathematical Physics. Some misprints have been eliminated

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