Week 10

1. Week 10 1. The Dirac delta function (continued) The delta function can be defined using several different families of functions. Introduce, for example, (1) and consider...

2. where f(x) is analytic in a certain interval about the point x = 0. It can be shown that i.e. the family of functions defined by (1) correspond to the delta function, just like the family introduced in Week 9. Theorem 1: A family of functions δε(x) such that

3. correspond to the delta function, i.e. (2) for any f(x) such that the integral in (2) converges for sufficiently small ε.

4. Example 1: Let For which a this family of functions corresponds to δε(x)?

5. 2. Derivatives of the Dirac delta function Differentiate the function δε(x) defined by (1), (3) Note that the values of δ'ε(x) at x = –ε, 0, +ε were chosen, more or less, ad hoc. Now, consider...

6. where f(x) is analytic in a certain interval about x = 0. Theorem 2: Proof: Represent f(x) by its Taylor series Observe that δ'ε(x) is odd – hence, every other term of the series in [] doesn’t contribute to the integral, and we obtain...

7. hence, hence, as required. █ tends to 0 asε→ 0

8. The ‘short-hand’ form of Theorem 2 is The minus on the r.-h.s. looks ‘unexpected’, but it actually makes sense. To see why, consider the definition of the delta function and replace f(x) in it with f'(x): Integrate the l.-h.s. by parts and assume that which yields...

9. The sign in this equality does agree with that in Theorem 2. Just like the delta function, δ'(x) can be defined using more than one family of functions. Instead of (3), one can pick any family δ'ε(x) such that

10. Theorem 3: The n-th derivative of δ(x) [denoted by δ(n)(x)] can be defined through any family of functions such that

11. We shall also use δ(x – x0) and δ'(x – x0), such that What’s the equivalent of the above equalities for δ"(x – x0)?

12. Theorem 4: Consider a function g(x) be strictly monotonic (g'(x) ≠ 0), such that (Give an example of such a function.) Such functions always have one and only one zero, which we shall denote x = x0, i.e. (Explain where the second condition comes from.) Then,

13. Proof: Consider and change the variable from x toy = g(x). Letting first g'(x0) > 0, we obtain hence, Since the function x(y) has the property x(0) = x0...

14. Recalling the definition of I, we have and the theorem is proven for the case g'(x0) > 0. The case g'(x0) < 0 is similar.

15. 3. Formal definition of generalised functions (GFs) We’ll formalise GFs by putting them in correspondence to linear functionals defined for some unproblematic (well-behaved) test functions. ۞An functionalF assigns to a function f(x) a number (which we shall denote [F, f(x)]). Example 2: (a) is a linear functional, (b) is a nonlinear functional.

16. ۞A function f(x) such that where a < b, is said to have compact support. ۞The space T of all analytic functions with compact support is called “the space of test functions”. Example 3: an analytic function with compact support where a < b.

17. ۞ A generalised function is a linear functional defined on T. Example 4: (a) The delta function is a linear functional (and, thus, a GF): (b) A ‘regular’ function g(x) can be treated as a GF: the test function the functional the actual function To avoid confusion, distinguish in the above the functionalG and the corresponding functiong(x)[and the test function f(x)].

18. ۞ The derivative of a GF G is the GF G' such that Theorem 5: If a ‘regular’ function g(x) corresponds to a generalised function G, then g'(x) corresponds to G'. Proof: By definition, Integrating the r.-h.s. by parts and recalling that f(x) is a function with compact support, we obtain...

19. as required. █ Example 5: a GF unrelated to the delta function Consider the GF corresponding to the following functional: For example,

20. Example 6: the Sokhotsky formula where the GF on the l.-h.s. is defined by