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1 Week 10 Generalised functions, or distributions 1.The Dirac delta function 2.Derivatives of the Dirac delta function 3.Differential equations involving generalised functions Consider a family of functions δ ε (x), where x is the variable and ε is a parameter: (1) 1. The Dirac delta function

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2 where f(x) is analytic in a certain interval about the point x = 0. (3) (2) Theorem 1: Consider Proof: Represent f(x) by its Taylor series: (4) Then

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3 hence, which yields (3) as required. █ Substitute (1) and (4) into (2): tends to 0 as ε → 0

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4 Consider (2) with f(x) = 1, which yields At the same time, it follows from (1) that Q: what kind of function tends to zero at all points except one, but still has non-zero integral (area under the curve)? A: It’s called the Dirac delta function, or just delta function. It’s not a usual function though, but a generalised function, or distribution.

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5 The delta function [usually denoted by δ(x) ] can be defined using infinitely many different families of functions. and consider... Introduce, for example, (5)

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6 It can be shown that i.e. the family of functions defined by (5) correspond to the delta function, just like family (1). where f(x) is analytic in a certain interval about the point x = 0.

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7 Then δ ε (x) corresponds to the delta function, i.e. for any f(x) which is analytic at x = 0. Let a family of functions δ ε (x) satisfy Theorem 2:

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8 Let Example 1: For which a does this family of functions correspond to δ(x) ?

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9 Even though generalised functions (GFs) imply an underlying limiting procedure, one often uses a ‘short-hand notation’ treating them as if they were regular functions, e.g. One should keep in mind, however, that the above equality actually means where δ ε (x) is a suitably defined family of functions. Comment: Yet, in many cases, the ‘short-hand notation’ can be used to re- arrange expressions involving GFs, and it yields the correct result!

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10 Example 2: where δ'(x) is the delta function’s derivative (we haven’t defined it yet, but let’s consider it anyway and see what happens). Treating δ(x) as a regular function and δ'(x) as its derivative, we integrate (6) by parts... Consider (6)

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11 Assume that Thus, which kind of agrees with the fact that, in the proper definition of δ(x), the function δ ε (x) vanishes outside the interval (–ε, ε). Now, recall how δ(x) affects test functions. Recalling also definition (6) of I, we obtain...

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12 Even though this equality was derived without following the proper procedure (families of functions, etc.), we’ll later see that (7) correctly describes how the derivative of the delta function affects a test function. (7)

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13 2. Derivatives of the Dirac delta function Consider the following family of functions: Note that, everywhere except the points x = –ε, 0, +ε, the function δ' ε (x) equals to the derivative of δ ε (x) defined by (5). At the ‘exceptional’ points, δ ε (x) doesn’t have a derivative, so the values of δ' ε (x) were chosen, more or less, ad hoc. Now, consider...

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14 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... Theorem 3: where f(x) is analytic in a certain interval about x = 0.

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15 hence, as required. █ tends to 0 as ε → 0 hence,

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16 The minus on the r.-h.s. looks ‘unnatural’, but it actually agrees with the result in Example 2 obtained using the short-hand notation. The ‘short-hand’ form of Theorem 3 is Comment: The family of functions used in Theorem 3 for representing δ'(x) was obtained by differentiating the family of functions defined by (5) and used to represent δ(x). In principle, we could’ve used a different family, but there’s still a general rule: if δ ε (x) represents δ(x), then the derivative of δ ε (x) represents δ'(x). Comment:

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17 The n -th derivative of δ(x) [denoted by δ (n) (x) ] can be defined through any family of functions such that Theorem 4:

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18 We shall also use δ(x – x 0 ) and δ'(x – x 0 ), such that What’s the equivalent of the above equalities for δ"(x – x 0 ) ?

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19 Theorem 5: Let a function g(x) be smooth and strictly monotonic, i.e. Then, Let also g(x) have a single zero at x = x 0, i.e. Note that, since g(x) is strictly monotonic,

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20 Proof: Let g'(x 0 ) > 0 and consider Let’s change the variable x to y = g(x), so that (8) where x(y) is the inverse function to y = g(x). Observe also that, since y(x 0 ) = 0 and x(y) is the inverse function, it follows that (9)

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21 Now, (8) becomes which can be readily evaluated using the definition of δ(x) : Taking into account (9), we obtain

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22 Finally, recalling definition (8) of I, we have which yields the desired result for the case g'(x 0 ) > 0. The case g'(x 0 ) < 0 is similar. █

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