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Lecture Notes on Dirac Delta Function, Fourier Transform, Laplace Transform

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Lecture Notes on Dirac Delta Function, Fourier Transform, Laplace Transform Lecture Notes on Dirac delta function, Fourier transform, Laplace transform Luca Salasnich Dipartment of Physics and Astronomy “Galileo Gailei” University of Padua 2 Contents 1 Dirac Delta Function 1 2 Fourier Transform 5 3 Laplace Transform 11 3 4 CONTENTS Chapter 1 Dirac Delta Function In 1880 the self-taught electrical scientist Oliver Heaviside introduced the following function 1 for x> 0 Θ(x)= (1.1) 0 for x< 0 which is now called Heaviside step function. This is a discontinous function, with a discon- tinuity of first kind (jump) at x = 0, which is often used in the context of the analysis of electric signals. Moreover, it is important to stress that the Haviside step function appears also in the context of quantum statistical physics. In fact, the Fermi-Dirac function (or Fermi-Dirac distribution) 1 F (x)= , (1.2) β eβx +1 proposed in 1926 by Enrico Fermi and Paul Dirac to describe the quantum statistical distribution of electrons in metals, where β = 1/(kBT ) is the inverse of the absolute temperature T (with kB the Boltzmann constant) and x = ǫ µ is the energy ǫ of the electron with respect to the chemical potential µ, becomes the function− Θ( x) in the limit − of very small temperature T , namely 0 for x> 0 lim Fβ(x)=Θ( x)= . (1.3) β→+∞ − 1 for x< 0 Inspired by the work of Heaviside, with the purpose of describing an extremely localized charge density, in 1930 Paul Dirac investigated the following “function” + for x =0 δ(x)= (1.4) 0∞ for x =0 6 imposing that +∞ δ(x) dx =1 . (1.5) Z−∞ Unfortunately, this property of δ(x) is not compatible with the definition (1.4). In fact, from Eq. (1.4) it follows that the integral must be equal to zero. In other words, it does 1 2 CHAPTER 1. DIRAC DELTA FUNCTION not exist a function δ(x) which satisfies both Eq. (1.4) and Eq. (1.5). Dirac suggested that a way to circumvent this problem is to interpret the integral of Eq. (1.5) as +∞ +∞ δ(x) dx = lim δǫ(x) dx , (1.6) ǫ→0+ Z−∞ Z−∞ where δǫ(x) is a generic function of both x and ǫ such that + for x =0 lim δǫ(x) = ∞ , (1.7) ǫ→0+ 0 for x =0 +∞ 6 δǫ(x) dx = 1 . (1.8) Z−∞ Thus, the Dirac delta function δ(x) is a “generalized function” (but, strictly-speaking, not a function) which satisfy Eqs. (1.4) and (1.5) with the caveat that the integral in Eq. (1.5) must be interpreted according to Eq. (1.6) where the functions δǫ(x) satisfy Eqs. (1.7) and (1.8). There are infinite functions δǫ(x) which satisfy Eqs. (1.7) and (1.8). Among them there is, for instance, the following Gaussian 1 2 2 δ (x)= e−x /ǫ , (1.9) ǫ ǫ√π which clearly satisfies Eq. (1.7) and whose integral is equal to 1 for any value of ǫ. Another example is the function 1 for x ǫ/2 δ (x)= ǫ , (1.10) ǫ 0 for |x| ≤> ǫ/2 | | which again satisfies Eq. (1.7) and whose integral is equal to 1 for any value of ǫ. In the following we shall use Eq. (1.10) to study the properties of the Dirac delta function. According to the approach of Dirac, the integral involving δ(x) must be interpreted as the limit of the corresponding integral involving δǫ(x), namely +∞ +∞ δ(x) f(x) dx = lim δǫ(x) f(x) dx , (1.11) ǫ→0+ Z−∞ Z−∞ for any function f(x). It is then easy to prove that +∞ δ(x) f(x) dx = f(0) . (1.12) Z−∞ by using Eq. (1.10) and the mean value theorem. Similarly one finds +∞ δ(x c) f(x) dx = f(c) . (1.13) − Z−∞ 3 Several other properties of the Dirac delta function δ(x) follow from its definition. In particular δ( x) = δ(x) , (1.14) − 1 δ(a x) = δ(x) with a =0 , (1.15) a 6 | | 1 δ(f(x)) = δ(x x ) with f(x )=0 . (1.16) f ′(x ) − i i i i X | | Up to now we have considered the Dirac delta function δ(x) with only one variable x. It is not difficult to define a Dirac delta function δ(D)(r) in the case of a D-dimensional domain RD, where r =(x , x , ..., x ) RD is a D-dimensional vector: 1 2 D ∈ + for r = 0 δ(D)(r)= ∞ (1.17) 0 for r = 0 6 and δ(D)(r) dDr =1 . (1.18) RD Z Notice that sometimes δ(D)(r) is written using the simpler notation δ(r). Clearly, also in this case one must interpret the integral of Eq. (1.18) as (D) D (D) D δ (r) d r = lim δǫ (r) d r , (1.19) RD ǫ→0+ RD Z Z (D) where δǫ (r) is a generic function of both r and ǫ such that (D) + for r = 0 lim δǫ (r) = ∞ , (1.20) ǫ→0+ 0 for r = 0 6 (D) D lim δǫ (r) d r = 1 . (1.21) ǫ→0+ Z Several properties of δ(x) remain valid also for δ(D)(r). Nevertheless, some properties of δ(D)(r) depend on the space dimension D. For instance, one can prove the remarkable formula 1 2 (ln r ) for D =2 (D) 2π ∇ | | δ (r)= 1 2 1 , (1.22) D−2 for D 3 ( − D(D−2)VD ∇ |r| ≥ 2 ∂2 ∂2 ∂2 D/2 where = 2 + 2 +...+ 2 and VD = π /Γ(1+D/2) is the volume of a D-dimensional ∇ ∂x1 ∂x2 ∂xD ipersphere of unitary radius, with Γ(x) the Euler Gamma function. In the case D = 3 the previous formula becomes 1 1 δ(3)(r)= 2 , (1.23) −4π ∇ r | | which can be used to transform the Gauss law of electromagnetism from its integral form to its differential form. 4 CHAPTER 1. DIRAC DELTA FUNCTION Chapter 2 Fourier Transform It was known from the times of Archimedes that, in some cases, the infinite sum of decreas- ing numbers can produce a finite result. But it was only in 1593 that the mathematician Francois Viete gave the first example of a function, f(x)=1/(1 x), written as the infinite sum of power functions. This function is nothing else than the− geometric series, given by 1 ∞ = xn , for x < 1 . (2.1) 1 x | | n=0 − X In 1714 Brook Taylor suggested that any real function f(x) which is infinitely differen- tiable in x0 and sufficiently regular can be written as a series of powers, i.e. ∞ f(x)= c (x x )n , (2.2) n − 0 n=0 X where the coefficients cn are given by 1 c = f (n)(x ) , (2.3) n n! 0 with f (n)(x) the n-th derivative of the function f(x). The series (2.2) is now called Taylor series and becomes the so-called Maclaurin series if x0 = 0. Clearly, the geometric series (2.1) is nothing else than the Maclaurin series, where cn = 1. We observe that it is quite easy to prove the Taylor series: it is sufficient to suppose that Eq. (2.2) is valid and then to derive the coefficients cn by calculating the derivatives of f(x) at x = x0; in this way one gets Eq. (2.3). In 1807 Jean Baptiste Joseph Fourier, who was interested on wave propagation and periodic phenomena, found that any sufficiently regular real function function f(x) which is periodic, i.e. such that f(x + L)= f(x) , (2.4) where L is the periodicity, can be written as the infinite sum of sinusoidal functions, namely a ∞ 2π 2π f(x)= 0 + a cos n x + b sin n x , (2.5) 2 n L n L n=1 X 5 6 CHAPTER 2. FOURIER TRANSFORM where 2 L/2 2π a = f(y) cos n y dy , (2.6) n L L Z−L/2 2 L/2 2π b = f(y) sin n y dy . (2.7) n L L Z−L/2 It is quite easy to prove also the series (2.5), which is now called Fourier series. In fact, it is sufficient to suppose that Eq. (2.5) is valid and then to derive the coefficients an and 2π 2π bn by multiplying both side of Eq. (2.5) by cos n L x and cos n L x respectively and integrating over one period L; in this way one gets Eqs. (2.6) and (2.7). It is important to stress that, in general, the real variable x of the function f(x) can represent a space coordinate but also a time coordinate. In the former case L gives the spatial periodicity and 2π/L is the wavenumber, while in the latter case L is the time periodicity and 2π/L the angular frequency. Taking into account the Euler formula 2π 2π 2π ein L x = cos n x + i sin n x (2.8) L L with i = √ 1 the imaginary unit, Fourier observed that his series (2.5) can be re-written in the very− elegant form +∞ 2π in L x f(x)= fn e , (2.9) n=−∞ X where L/2 1 2π f = f(y) e−in L y dy (2.10) n L Z−L/2 are complex coefficients, with f0 = a0/2, fn =(an ibn)/2 if n> 0 and fn =(a−n +ib−n)/2 ∗ − if n< 0, thus fn = f−n.
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