Treffer: Series and integral representations for the biregular exponential function

The Cauchy-Kowalewski extension principle relates a unique left monogenic extension to any real-analytic function. First, this principle is used as starting point for surveying possible definitions (and extensions) of biregular exponential functions and related results in connection with monogenic functions, Fourier transforms, and hyperfunctions. In view of an article by the second author [Rend. Circ. Mat. Palermo, II. Ser., Suppl. 9, 205-219 (1985; Zbl 0597.30059)], the paper under review has two aims, namely, (i) to establish a series expansion for the biregular extension of the exponential function \[ \exp i\langle\underline x,\underline t\rangle\;\Biggl(\langle \underline x,\underline t\rangle= \sum^m_{j=1} x_j t_j\Biggr) \] based on the Fischer decomposition, and (ii) to present a double integral formula leading to a four-fold decomposition thus generalizing an analogous decomposition result of the monogenic exponential.