A Complete Characterization of the Derivative of a Polygenic Function

generally leads to a functional differential equation for f(x) and an associated inequality. It is only occasionally that the equation forf(x) reduces to a differential equation but it may sometimes reduce to a simple functional equation or difference equation. The analysis is easily extended by replacing the substitution z = x + r by some other substitution z = s(x) which transforms the range of integration into itself by a one to one correspondence.** The equation derived from the variation problem is then an iterative differential equation involving functions of x, s(x) and s(s(x)). * This condition may be replaced by some other condition such as 2wf(x)g(x)dx = 1. ** Another generalization is obtained by considering a summation over all integral values of n of p,[u,,fn + vj,,+ + e.I2, when the sum of f is unity,and Pn > 0.