When functions have no value(s): Delta functions and distributions

x = 0. That is, one would like the function δ(x) = 0 for all x 6 = 0, but with δ(x)dx = 1 for any inThese notes give a brief introduction to the motegration region that includes ́ x = 0; this concept tivations, concepts, and properties of distributions, is called a “Dirac delta function” or simply a “delta which generalize the notion of functions f(x) to alfunction.” δ(x) is usually the simplest right-handlow derivatives of discontinuities, “delta” functions, side for which to solve differential equations, yielding and other nice things. This generalization is ina Green’s function. It is also the simplest way to creasingly important the more you work with linear consider physical effects that are concentrated within PDEs, as we do in 18.303. For example, Green’s funcvery small volumes or times, for which you don’t actions are extremely cumbersome if one does not altually want to worry about the microscopic details low delta functions. Moreover, solving PDEs with in this volume—for example, think of the concepts of functions that are not classically differentiable is of a “point charge,” a “point mass,” a force plucking a great practical importance (e.g. a plucked string with string at “one point,” a “kick” that “suddenly” imparts a triangle shape is not twice differentiable, making some momentum to an object, and so on. The probthe wave equation problematic with traditional funclem is that there is no classical function δ(x) having tions). Any serious work with PDEs will eventually these properties. run into the concept of a “weak solution,” which is For example, one could imagine constructing this essentially a version of the distribution concept. function as the limit: { 1 1 What’s wrong with functions? δ(x) = lim ∆x→0+ The most familiar notion of a function f(x) is a map from real numbers R to real numbers R (or maybe complex numbers C); that is, for every x you have a value y = f(x). Of course, you may know that one can define functions from any set to any other set, but at first glance it seems that R → R functions (and multi-dimensional generalizations thereof) are the best-suited concept for describing most physical quantities—for example, velocity as a function of time, or pressure or density as a function of position in a fluid, and so on. Unfortunately, such functions have some severe drawbacks that, eventually, lead them to be replaced in some contexts by another concept: distributions (also called generalized functions). What are these drawbacks? 1.1 No delta functions For lots of applications, such as those involving PDEs and Green’s functions, one would like to have a function δ(x) whose integral is “concentrated” at the point ∆x 0 ≤ x < ∆x 0 otherwise = lim ∆x→0+ δ∆x(x)? For any ∆x > 0, the function δ∆x(x) at right has integral = 1 and is zero except near x = 0. Unfortunately, the ∆x → 0 limit does not exist as an ordinary function: δ∆x(x) approaches ∞ for x = 0, but of course ∞ is not a real number. Informally, one often sees “definitions” of δ(x) that describe it as some mysterious object that is “not quite” a function, which = 0 for x 6= 0 but is undefined at x = 0, and which is “only really defined inside an integral” (where it “integrates” to 1).1 This may leave you with a queasy feeling that δ(x) is somehow 1Historically, this unsatisfactory description is precisely how δ(x) first appeared, and for many years there was a corresponding cloud of guilt and uncertainty surrounding any usage of it. Most famously, an informal δ(x) notion was popularized by physicist Paul Dirac, who in his Principles of Quantum Mechanics (1930) wrote: “Thus δ(x) is not a quantity which can be generally used in mathematical analysis like an ordinary function, but its use must be confined to certain simple types of expression for which it is obvious that no inconsistency can arise.” You know you are on shaky ground, in mathematics, when you are forced to appeal to the “obvious.”