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




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Abstract: We present a generalization of the representation in plane waves of Diracdelta, $\deltax=1-2\pi\int {-\infty}^\infty e^{-ikx}\,dk$, namely$\deltax=2-q-2\pi\int {-\infty}^\infty e q^{-ikx}\,dk$, using thenonextensive-statistical-mechanics $q$-exponential function,$e q^{ix}\equiv1+1-qix^{1-1-q}$ 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 transcendentalnumber $\pi$. Incidentally, we remark that the $q$-plane wave form whichemerges, namely $e q^{ikx}$, is normalizable for $1


Author: M. Jauregui, C. Tsallis

Source: https://arxiv.org/



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