The Cauchy-Riemann Differential Equations of Complex Functions

The articles [20], [21], [6], [7], [22], [8], [3], [1], [4], [14], [13], [19], [16], [9], [2], [5], [10], [17], [11], [18], [12], and [15] provide the notation and terminology for this paper. Let f be a partial function from C to C. The functor <(f) yielding a partial function from C to R is defined as follows: (Def. 1) dom f = dom<(f) and for every complex number z such that z ∈ dom<(f) holds <(f)(z) = <(fz). Let f be a partial function from C to C. The functor =(f) yields a partial function from C to R and is defined as follows: (Def. 2) dom f = dom=(f) and for every complex number z such that z ∈ dom=(f) holds =(f)(z) = =(fz). One can prove the following propositions: (1) For every partial function f from C to C such that f is total holds dom<(f) = C and dom=(f) = C.