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

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

2010-03-25

arxiv.org/abs/1003.4967v3

J. Math. Phys. 51, 063304 (2010)

Physics

Mathematical Physics

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

Scientific paper

10.1063/1.3431981

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

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