Physics 215 Winter 2018 Examples of Generalized Functions The following results are based on material from I.M. Gel’fand and G.E. Shilov, Gen- eralized Functions, Volume 1: Properties and Operations (Academic Press, New York, NY, 1964), and D.S. Jones, The Theory of Generalised Functions, 2nd edition (Cambridge Uni- versity Press, Cambridge, UK, 1982). We begin by introducing the Heavyside step function, 1 , for x> 0 , 0 , for x> 0 , Θ(x)= and Θ(−x) ≡ 1 − Θ(x)= , (1) (0 , for x< 0 . (1 , for x< 0 . A specific value of Θ(x) at x = 0 is not specified. The delta function can then be defined as d δ(x)= Θ(x) . dx The delta function δ(x) is not a function at all; instead it is a generalized function that only makes formal sense when first multiplied by a function f(x) that is smooth and non-singular in a neighborhood of the origin, and then integrated over a range of x containing the origin. We shall also assume that f(x) → 0 sufficiently fast as x → ±∞ in order that integrals evaluated over the entire real line are convergent. It then follows that surface terms at x = ±∞, which arise when integrating by parts, vanish. The allowed set of functions f(x) forms the space of test functions. In the case of the delta function, we have ∞ ∞ ∞ ∞ dΘ ∞ df δ(x)f(x) dx = f(x) dx = f(x)Θ(x) − Θ(x) dx = −f(x) = f(0) . −∞ −∞ dx −∞ dx Z Z −∞ Z 0 By a similar computation, one can verify that the generalized function, xδ(x) = 0. The generalized function, P(1/x), can be defined by the following equation, ∞ f(x) ∞ f(x) − f(−x) P dx = dx , (2) −∞ x x Z Z0 where f(x) is regular in a neighborhood of the real axis and vanishes as |x|→∞. In the Appendix, we show that eq. (2) is equivalent to the definition of the Cauchy principal value, ∞ f(x) dx −δ f(x) dx ∞ f(x) dx P ≡ lim + , (3) −∞ x δ→0 −∞ x x Z Z Zδ −1 −1 where δ > 0. Gel’fand and Shilov define two additional functions, x+ and x− via −1 −1 1 x , for x> 0 , x+ = Θ(x) = (4) x (0 , for x< 0 . −1 1 0 , for x> 0 , x− = −Θ(−x) = −1 (5) x (|x| , for x< 0 . 1 −1 −1 As in the case of P(1/x), we would like to extend the definition of x+ and x− such that they yield finite results when integrated against a test function over the real axis. The corresponding generalized functions are defined by,1 1 1 1 = lim Θ(x) 1−µ − δ(x) , (6) x+ µ→0 x µ 1 1 1 = lim Θ(−x) 1− − δ(x) , (7) x− µ→0 (−x) µ µ where µ is a real and positive infinitesimal. It then follows that ∞ f(x) ∞ f(x) 1 ∞ dx = lim 1−µ dx − δ(x)f(x) dx −∞ x+ µ→0 x µ −∞ Z Z0 Z 1 f(x) 1 ∞ f(x) = lim − dx − f(0) + dx µ→0 x1 µ µ x Z0 Z1 1 f(x) − f(0) ∞ f(x) 1 dx 1 = dx + dx + lim f(0) − dx − f(0) .(8) x x µ→0 x1 µ µ Z0 Z1 Z0 In the last step above, we wrote f(x) = f(x) − f(0) + f(0) and took the µ → 0 limit in the first term, which is allowed since the corresponding integral converges for any smooth function f(x) that vanishes as x → ±∞. Finally, the last two terms in eq. (8) cancel exactly, and we end up with a well-defined and finite result, ∞ f(x) ∞ f(x) − f(0)Θ(1 − x) 1 f(x) − f(0) ∞ f(x) dx = dx = dx + dx . (9) −∞ x+ x x x Z Z0 Z0 Z1 A similar computation yields, ∞ f(x) ∞ f(−x) − f(0)Θ(1 − x) 1 f(−x) − f(0) ∞ f(−x) dx = dx = dx + dx . −∞ x− 0 x 0 x 1 x Z Z Z Z (10) It is easy to verify that the generalized functions 1/x± satisfy, 1 1 1 P = − . (11) x x+ x− Note that eq. (11) is true for x =6 0 in light of the definitions of x± given by eqs. (4) and (5). In addition, one can check that subtracting eq. (10) from eq. (9) yields eq. (2). One can also use eq. (11) to obtain another definition of P(1/x) by employing eqs. (6) and (7), 1 1 1 P = lim Θ(x) − − Θ(−x) − . (12) x µ→0 x1 µ (−x)1 µ 1 −1 −1 The conditions, xx+ = Θ(x) and xx− = −Θ(−x), do not yield unique generalized functions. For −1 example, x x+ + Cδ(x) = Θ(x) for any constant C. The definitions given in eqs. (6) and (7) are taken from D.S. Jones, The Theory of Generalised Functions, op. cit., and are motivated by the desire that the integrals given in eqs. (9) and (10) should be well-defined and finite. Some books write Pf(1/x±) in eqs. (6) and (7) to distinguish these generalized functions from the ones defined in eqs. (4) and (5) [where Pf stands for pseudofunction], but following Gel’fand and Shilov (and Jones) we will not do so in these notes. 2 In light of |x| = −x for x< 0, it follows that 1 1 1 sgn(x) Θ(x) − Θ(−x) = Θ(x) − Θ(−x) = , (13) x1−µ (−x)1−µ |x|1−µ |x|1−µ where sgn(x) is the sign function, +1 , for x> 0, sgn(x)=Θ(x) − Θ(−x)= (14) (−1 , for x< 0. Hence, eq. (12) yields 1 sgn(x) P = lim − . (15) x µ→0 |x|1 µ It is a simple exercise to verify that multiplying the right hand side of eq. (15) by f(x) and integrating over the real line yields eq. (2), f(x) ∞ f(x) ∞ f(x) 0 f(x) P dx = lim 1−µ sgn(x) dx = lim 1−µ dx − 1−µ dx x µ→0 −∞ |x| µ→0 x −∞ x Z Z Z0 Z ∞ f(x) − f(−x) ∞ f(x) − f(−x) = lim − dx = dx , (16) µ→0 x1 µ x Z0 Z0 after changing the integration variable x → −x in the second integral in the penultimate step above. In the final step, we can set µ = 0 since the resulting integral is well-defined and finite, under the assumption that f(x) is a smooth function that vanishes as x → ±∞. Next, we consider the function, 1 1 x−1 , for x> 0 , = Θ(x) − Θ(−x) = −1 (17) |x| x (−x , for x< 0 . We again propose to extend the definition of |x|−1 such that it yields a finite result when integrated against a test function over the real axis. The corresponding generalized function, denoted by Pf(1/|x|), is defined by,2 1 1 2 Pf = lim − − δ(x) . (18) |x| µ→0 |x|1 µ µ −1 In order to see the relation between Pf(1/|x|) and the generalized functions x± , we employ Θ(x)+Θ(−x) = 1 [cf. eq. (1)] to obtain the identity, 1 1 1 1 = Θ(x)+Θ(−x) = Θ(x) + Θ(−x) . (19) |x|1−µ |x|1−µ x1−µ (−x)1−µ 2A generalized function that coincides with 1/|x| for x 6= 0 not unique; see footnote 1. One particularly convenient choice is Pf(1/|x|), which in the notation of Gel’fand and Shilov (and Jones) is denoted as1/|x|. However, in these notes, we prefer to keep the Pf (pseudofunction) symbol in this case. 3 Hence, one can re-express eq. (18) as, 1 1 1 2 Pf = lim Θ(x) − + Θ(−x) − − δ(x) . (20) |x| µ→0 x1 µ (−x)1 µ µ Comparing with eqs. (6) and (7), we see that 1 1 1 Pf = + . (21) |x| x+ x− −1 −1 Note that eq. (21) is clearly true for x =6 0 in light of the initial definitions of x± and |x| given by eqs. (4), (5) and (17). It is convenient to rewrite eq. (10) by changing the integration variable x → −x and writing |x| = −x for x< 0, which yields ∞ f(x) 0 f(x) − f(0)Θ(1 + x) −1 f(x) 0 f(x) − f(0) dx = − dx = dx + dx . (22) −∞ x− −∞ x −∞ |x| − |x| Z Z Z Z 1 Using eqs. (9) and (22), it then follows that ∞ 1 −1 f(x) 1 f(x) − f(0) ∞ f(x) f(x) Pf dx ≡ dx + dk + dx , (23) −∞ |x| −∞ |x| − |x| |x| Z Z Z 1 Z1 which is a well-defined and finite result, assuming f(x) is smooth and vanishes as x → ±∞. A few more generalized functions are of interest. First, limε→0 1/(x ± iε) is related to other known generalized functions via the Sokhotski-Plemelj formula, 1 1 lim = P ∓ iπδ(x) , (24) ε→0 x ± iε x where ε> 0 is an infinitesimal real quantity. Perhaps less well known is the related formula,3 1 1 1 1 2 lim ln(x ± iε) = P ln |x| − ∓iπ + 2 π δ(x) , (25) ε→0 x ± iε x x− In eq. (25), x is a real variable and the logarithm is given by its principal value, lim ln(x ± iε) = ln |x| ± iπΘ(−x) . (26) ε→0 To derive eq. (25), we first take the square of eq.