# Complex-dimensional Integral and Light-cone Singularities

### Noboru Nakanishi

Kyoto University, Japan

## Abstract

The notion of a complex-dimensional integral is introduced in the complex *n*-dimensional Minkowski space. Its basic properties, such as Lorentz invariance, are investigated. Complex-dimensional invariant delta functions *Δn*(_x,m_2), *Δ(1)n*(_x,m_2), etc. are explicitly calculated in position space. It is proposed to define products of singular functions in the ordinary Minkowski space by analytically continuing the corresponding *n*-dimensional ones to *n*=4. The light-cone singularities of [*Δ*(_x,m_2)]2, *Δ*(_x,m_2) x *Δ*(1)(_x,m_2)]2 and *Δ*(1)(_x,m_2)]2 are shown to be unambiguously determined in this way.

## Cite this article

Noboru Nakanishi, Complex-dimensional Integral and Light-cone Singularities. Publ. Res. Inst. Math. Sci. 12 (1976), no. 99, pp. 343–346

DOI 10.2977/PRIMS/1195196612