How can we extend the idea of derivative so that more functions are differentiable? Why would we want to do so? How can we make sense of a delta “function” that isn’t really a function? We’ll answer these questions in this post.
Suppose f(x) is a differentiable function of one variable. Suppose φ(x) is an infinitely differentiable function that is zero outside of some finite interval. Functions like φ are called test functions. Integration by parts says that
where the integrals are over the entire real line. (The fact that φ is zero outside a finite interval mean the “uv” term from integration by parts is zero.) Now suppose f(x) is not differentiable. Then the left side of the equation above does not make sense, but the right side does. We use the right hand side to develop the definition of the generalized derivative.
We think of the function f not as a function of real numbers, but as a distribution that operates on tests functions. That is, we associate with f the linear functional on the space of tests functions that maps φ to ∫ f(x) φ(x) dx. Then the distributional derivative of this functional is another linear functional, the distribution that maps test functions φ to -∫ f(x) φ'(x) dx. In summary,
We can use this procedure to define as many derivatives of f as we’d like, as long as f is integrable. So f could be some horribly ill-behaved function, differentiable nowhere in the classical sense, and we could define its 37th derivative by repeatedly applying this idea. (Distributions are also called “generalized functions.” Distributional derivatives are also called “generalized derivatives” or “weak derivatives.”)
By the way, this same procedure is used to make sense of the delta function. The delta function isn’t a function at all. It is the distribution δ that evaluates test functions at zero, i.e. δ maps φ to φ(0). (The delta function often nonsensically defined to be a function that is infinite at zero and zero everywhere else.)
Why would we want to be able to differentiate more functions? When we can differentiate more functions, we can look in a bigger space for solutions to differential equations. Sometimes this allows us to find solutions to equations that do not have classical solutions. Other times this allows us to find classical solutions more easily. We may first prove that a generalized solution exists, and then prove that the generalized solution is in fact a classical solution.
Here’s an analogy that explains how generalized solutions might lead to classical solutions. Suppose you want to find the minimum value of a function for integer arguments. You might first look for a real number that minimizes the function. This lets you, for example, use derivatives in your search for the minimum. If the real minimum you find happens to also be an integer, then you’ve solved your original problem. Distributions and generalized derivatives work much the same way. You might find a classical solution by first looking in a larger space of possible solutions, a space that allows you to use more powerful techniques in your search for a solution.