Commutative Extended Complex Numbers and Connected Trigonometry

In order to construct an extension of the complex numbers, we consider an n-dimensional commutative algebra generated by the n vectors 1, e, ..., en−1 where the fundamental element satisfies the basic relation en = −1. These spaces can be classified according to the values of n: prime number, power of a prime number, general number. The question of the invertibility leads to the definition of a pseudo-norm for which the triangle inequality is not satisfied (the n = 1, 2 cases excepted). When one tries to pass from the polar form the cartesian one, one obtains functions generalizing the usual circular and hyperbolic functions and their inverse. The extended sine and hyperbolic sine functions thus constructed satisfy a determinantal-type relation and they lay the foundation of a new trigonometry for which summation and derivative formulas are given. An extended 2π quantity is defined as the periodicity of the generalized circular functions. This formalism is applied to solve the nth order differential equations (∑n−1i=1 (∂n/∂φni) ± ω) ƒ(φ) = 0. As a further application, the solutions of the n-laplacian operator are derived.