用非广延统计力学q指数函数表示π和Dirac δ
New representations of pi and Dirac delta using the nonextensive-statistical-mechanics q-exponential function
摘要：
我们提出了一个广义的狄拉克三角洲，$δ(x)=(1/2π)\int_{-\infty}^\infty e^{-ikx}\,dk$，即$δ(x)=(2-q)/(2π)\int_{-\infty}^\infty e_q^{-ikx}\,dk$，在平面波的表示，使用非扩展统计力学$q$指数函数，$e_q^{ix}\equiv[1+(1-q)ix]^{1/(1-q)}$与$e_1^{ix}\equiv e^{ix}$，是$x$任何实数，为$q$的实值在区间$[1,2[$。伴随着这些新的狄拉克δ表示的发展，我们也提出了两个新的家庭的超越数PDMATHAMT表示。顺便说一句，我们注意到，出现的$q$平面波形，即$e_q^{ikx}$，与标准波形$e^{ikx}$相反，对于$1<q<3$是可归一化的。
英文摘要：
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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