Some new uses of the η m (Z) functions. We present a procedure and a MATHEMATICA code for the conversion of formulae expressed in terms of the trigonometric functions sin(ωx), cos(ωx) or hyperbolic functions sinh(λx), cosh(λx) to forms expressed in terms of η m (Z) functions. The possibility of such a conversion is important in the evaluation of the coefficients of the approximation rules derived in the frame of the exponential fitting. The converted expressions allow, among others, a full elimination of the 0/0 undeterminacy, uniform accuracy in the computation of the coefficients, and an extended area of validity for the corresponding approximation formulae.