A new continuous entropy function and its inverse

A new continuous entropy function of the form h(x)=-k∫p'(x)*ln(p(x)) is constructed based on Lebesgue integral. Solution of the integral is of the form h(x)=-k*p(x)*ln(e-1*p(x)). Inverse solution of this function in the form of p(x)=h(x)*W-1(h(x)* e-1) has been obtained and this is a novelty. The fact that the integral of a logarithmic function is also a logarithmic function has been exploited and used in a more general ansatz for a deliberate function. The solution of the differential equation Λ(x)= g(x)'*ln(g(x)) for a deliberate function g(x) includes the Lambert W function. The solution is a parametric function g(x)=(s(x)+C)/W(e-1*(s(x)+C)). Parameter C has a minimal value and, consequently, an infimal function g(x) exists. This transform is of general use, and is particularly suitable for use in neural networks, the measurement of complex systems and economic modeling, since it can transform multivariate variables to exponential form.