Representability by Differential Equations

As a third argument for the sufficiency of combed generalized functions for physics, we may go back to the fact that physics is often expressed by second-order differential equations. Given an arbitrary second-order differential equation to be satisfied by a real function $f(\theta)$, if we are to consider the derivatives involved in the standard way, then the function $f(\theta)$ must necessarily be twice-differentiable, and therefore differentiable and continuous. Since every continuous function is a combed function, this implies that $f(\theta)$ must be a combed function, and in this case also a normal real function. However, just as in the standard Schwartz theory of distributions, the introduction of generalized functions represented by inner analytic functions allows us to generalize the concept of differentiability and hence to enlarge the scope of the differential equations.

If a real function $f(\theta)$ is not strictly differentiable at a given singular point, we may still be able to uniquely attribute a value to its derivative at that point, by the following sequence of operations: first we go to the open unit disk of the complex plane and take the inner analytic function $w(z)$ associated to that real function; second, since the inner analytic function is always differentiable, we then take its logarithmic derivative, which produces another inner analytic function within the open unit disk; finally, we take the limit of the real part of this new complex analytic function to the singular point on the unit circle; it that limits exists (as it often will), we attribute that value to the derivative of the original function at the singular point.

This can be further generalized, in case the limit to the unit circle does not exist in a strict sense, by the global attribution of a singular generalized function as the derivative of a strictly non-differentiable function. In this way one may state, for example, that the derivative of a function with a step-type singularity of height one at a certain point contains a delta ``function'' centered at that point. From the point of view of this more general definition of the derivative, all combed generalized functions are differentiable, and in fact infinitely differentiable, just as in the standard Schwartz theory of distributions. We are therefore able to consider arbitrary generalized functions as possible solutions of differential equations, so long as we reinterpret these equations in terms of generalized derivatives.

Note that this extension of the scope of differential equations, if executed via the representation of the generalized functions by inner analytic functions, is done without leaving the realm of combed objects, which in this case are combed generalized functions. Therefore, once again we may argue that combed generalized functions suffice for the description of physics.