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.
Categories: math-ph cond-mat.stat-mech math.MP