Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Functions and Operators

At this point it seems appropriate to observe that Servois does not make a distinction between “function" and “operator" as we do today. Sometimes he uses the word “function" in the same way that students of calculus do (i.e., to describe a relation between independent and dependent variables), but he also uses it to describe operators, functions of functions such as the varied state, and difference operators. One must keep in mind that the formal definition of the term “function" would not be introduced until 1837 by Dirichlet [Katz 2009]. Once again, for the sake of clarity, we will often distinguish functions from operators in this guide.

Servois defines the varied state and difference for multivariable functions. Generally, Servois calls a multivariable function a “complex.”  However, these should not be confused with complex numbers.  On the other hand, in the one instance where Servois does consider complex quantities later in the “Essay,” he uses the word “imaginary,” which was the standard term at the time.

Early within the “Essay” we witness Servois establishing a sort of group structure for operators, defining the identity operator, \( f^0(z) \), and the inverse operator. He calls the inverse of his difference operator an integral, although “sum” might be more appropriate, and he uses the symbol \( \sum \) and not \( \int \). He observes that it takes an arbitrary additive complement, analogous to the constant of integration. From here Servois establishes some common inverse functions, such as \(\ln z\) and \(e^z\). Servois uses \(\mbox{L}\) to denote the natural logarithm; however, we will use the modern “\(\ln\)” notation in this guide.

Servois uses the term polynomial for any function or operator \(F\) of the form \[ F(z)=\mbox{f}(z) + f(z) + \varphi (z) + \ldots, \] where \(\mbox{f}\), \(f\), \(\varphi\), \(\ldots\) are the composing monomial functions. This literal use of the word “polynomial” is much broader than the modern use. Not only may the constituents be functions other than the familiar monomial functions \(ax^n\), they may be operators, including the partial varied state, which increments only a single variable in a multivariable function, or the partial operator that multiplies a single variable by a constant. Furthermore, later in the paper, Servois considers “polynomial” functions with an infinite number of constituents.