Equivalent Analytical Functions of Sums of Sigmoid like Transcendental Functions

There is no mathematical solution to adding up transcendental functions other than numerical process. This paper put forward analytical method to model the sum of sigmoid like functions with an equivalent function. The Brillouin and Langevin as well as the error function, the tanh, sigmoid and the tan^-1 functions are investigated, their equivalent functions are calculated for four components and the error between the numerical (computer assisted) result and the equivalent function is tested for accuracy. The best modelling function, the most useful to include into mathematical operations, is pointed out finally, based on its performance and convenience. The paper intends to help people involved mostly in modelling hysteresis in Magnetism and other field of research in physics.