The Hankel transform of causal distributions

Abstract

In this note we evaluate the unidimensional distributional Hankel transform of \dfrac{x^{\alpha-1}_{+}}{\Gamma^{\alpha}},\dfrac{x^{\alpha-1}_{-}}{\Gamma^{\alpha}},dfrac{|x|^{\alpha-1}}{\Gamma^{\frac{\alpha}{2}}},dfrac{|x|^{\alpha-1}sgn(x)}{\Gamma^{\frac{\alpha +1}{2}}} and (x± i0)^{\alpha-1} and then we extend the formulae to certain kinds of n-dimensional distributions calles "causal" and "anti-causal" distributions. We evaluate the distributional Handel transform of \dfrac{(m^2+P)^{\alpha -1}_{-}}{\Gamma^{(\alpha)} }, \dfrac{|m^2+P|^{\alpha -1}_{-}}{\Gamma^{(\frac{\alpha}{2})}}, \dfrac{|m^2+P|^{\alpha -1}sgn(m^2+P)}{\Gamma (\frac{\alpha +1}{2 })} and (m^2+P±i0)^{\alpha-1}In this note we evaluate the unidimensional distributional Hankel transform of \dfrac{x^{\alpha-1}_{+}}{\Gamma^{\alpha}},\dfrac{x^{\alpha-1}_{-}}{\Gamma^{\alpha}},dfrac{|x|^{\alpha-1}}{\Gamma^{\frac{\alpha}{2}}},dfrac{|x|^{\alpha-1}sgn(x)}{\Gamma^{\frac{\alpha +1}{2}}} and (x± i0)^{\alpha-1} and then we extend the formulae to certain kinds of n-dimensional distributions calles "causal" and "anti-causal" distributions. We evaluate the distributional Handel transform of \dfrac{(m^2+P)^{\alpha -1}_{-}}{\Gamma^{(\alpha)} }, \dfrac{|m^2+P|^{\alpha -1}_{-}}{\Gamma^{(\frac{\alpha}{2})}}, \dfrac{|m^2+P|^{\alpha -1}sgn(m^2+P)}{\Gamma (\frac{\alpha +1}{2 })} and (m^2+P±i0)^{\alpha-1}