Non-newtonian analysis generated by hyperbolic tangent function

Jane Grossman, Michael Grossman and Robert Katz first introduced sigmoidal arithmetic with the generator tanh(x/2) in 1983. They gave the addition and subtraction operations related to this arithmetic (Grossman et al., 1983). Marek Czachor gave a definition of the arithmetic of which the hyperbolic tangent function, tanh(x), is the generator and the related operations of addition, subtraction, multiplication and division (Czachor, 2021). But he did not define the derivative and integral of this analysis. In this thesis, a non-Newtonian analysis, which is called hyperbolic tangent analysis, where the hyperbolic tangent function (tanh(x)) is the generator, has been studied. Hyperbolic tangent derivative and hyperbolic tangent integral were introduced for the first time. Additionally, fundamental definitions and theorems related to this analysis are given.