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arXiv · 1003.4967

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

M. Jauregui·C. Tsallis
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Abstract

We present a generalization of the representation in plane waves of Dirac delta, $δ(x)=(1/2π)\int_{-\infty}^\infty e^{-ikx}\,dk$, namely $δ(x)=(2-q)/(2π)\int_{-\infty}^\infty e_q^{-ikx}\,dk$, 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 $π$. 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.

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M. Jauregui, C. Tsallis. 2010-05-24. New representations of pi and Dirac delta using the nonextensive-statistical-mechanics q-exponential function. https://doi.org/10.1063/1.3431981

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