A relation between the \(k\)th derivate of the Dirac delta in \((P\pm i0)\) and the residue of distributions \((P\pm i0)^\lambda\) (Q1807790)

Let \(x= (x_1,\dots, x_n)\in \mathbb{R}^n\) and let \(P= P(x)\) be a quadratic form \(x^2_1+\cdots+ x^2_p- x^2_{p+1}-\cdots- x^2_{p+q}\), where \(p+q= n\). If \(\lambda\) is a complex number and \(|x|^2= x^2_1+\cdots+ x^2_n\) the symbols \((P\pm i0)^\lambda\) stand for the distributions defined as \(\lim_{\varepsilon\to 0}(P\pm i\varepsilon|x|^2)^\lambda\) and considered as holomorphic distribution valued functions of \(\lambda\) everywhere except in points of the form \(\lambda=-{n\over 2}-k\), where \(k\) is a nonnegative integer. Such points are first-order poles of these distributions and for the residua \(\underset{\lambda=-{n\over 2}-k}{\text{res}}(P\pm 0)^\lambda\) the following equality has been found: \[ \underset{\lambda=-{n\over 2}-k} {\text{res}}(P\pm 0)^\lambda= {e^{\pm i({q\over 2})\pi}\pi^{{n\over 2}}\over 4^kk!\Gamma({n\over 2}+ k)} L^k\delta, \] where \(L={\partial^2\over\partial x^2_1}+\cdots+ {\partial^2\over\partial x^2_p}- {\partial^2\over\partial x^2_{p+ 1}}-\cdots- {\partial^2\over\partial x^2_{p+q}}\). As the main result, the authors prove the following statement: Let \(k\) be a nonnegative integer such that \(k\geq{n\over 2}\). Then if \(p\) and \(q\) are both even then \[ \underset{\lambda=-k-1} {\text{res}}(P\pm i0)^\lambda= {(-1)^k\over k!} \delta^{(k)}(P\pm i0), \] where \(\delta^{(k)}(P\pm i0)= \delta^{(k)}(P_+)+ e^{\pm\pi i(k+ 1)}\delta^{(k)}(P_-)\).