The Dirac Delta Function

Instituto de Ciencia y Tecnolog´ıade Materiales (IMRE)-Facultad de F´ısica Universidad de la Habana

IUCr International School on Crystallography, Brazil, 2012. III Latinoamerican series Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Outline

1 Introduction Deﬁning the Dirac Delta function

2 Dirac delta function as the limit of a family of functions

3 Properties of the Dirac delta function

4 Dirac delta function obtained from a complete set of orthonormal functions Dirac comb

5 Dirac delta in higher dimensional space

6 Recapitulation

7 Exercises

8 References

The Dirac Delta function 2 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. The Kronecker Delta

Suppose we have a sequence of values {a1, a2,...} and we wish to select algebraically a particular value labeled by its index i

The Dirac Delta function 3 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. The Kronecker Delta

Deﬁnition (Kronecker delta) 1 i = j K δ = ij 0 i 6= j

Picking one member of a set algebraically

X K aj δij = ai j=1

The Dirac Delta function 4 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. The Kronecker Delta

Deﬁnition (Kronecker delta) 1 i = j K δ = ij 0 i 6= j

Picking one member of a set algebraically

X K aj δij = ai j=1

The Dirac Delta function 4 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Properties of the Kronecker Delta

P K j δij = 1: Normalization condition.

K K δij = δji: Symmetry property.

The Dirac Delta function 5 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Properties of the Kronecker Delta

P K j δij = 1: Normalization condition.

K K δij = δji: Symmetry property.

The Dirac Delta function 5 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Spotting a point in the mountain proﬁle

We want to pick up just a narrow window of the whole view

The Dirac Delta function 6 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. How to deal with continuous functions ?

We want to do the same with a continuous function.

The Dirac Delta function 7 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Deﬁning the Dirac Delta function

Consider a function f(x) continuous in the interval (a, b) and suppose we want to pick up algebraically the value of f(x) at a particular point labeled by x0. In analogy with the Kronecker delta let us deﬁne a selector function Dδ(x) with the following two properties:

R b D a f(x) δ(x − x0)dx = f(x0): Selector or sifting property

R b D a δ(x − x0)dx = 1: Normalization condition

The Dirac Delta function 8 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Deﬁning the Dirac Delta function

R b D a f(x) δ(x − x0)dx = f(x0): Selector or sifting property

R b D a δ(x − x0)dx = 1: Normalization condition

The Dirac Delta function 8 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Deﬁning the Dirac Delta function

To be more general consider f(x) to be continuous in the interval (a, b) except in a finite number of points where finite discontinuities occurs, then the Dirac Delta can be defined as Definition (Dirac delta function)  1 [f(x−) + f(x+] x ∈ (a, b)  2 0 0 0 Z b  1 + 2 f(x0 ) x0 = a f(x)δ(x − x0)dx = 1 − f(x ) x0 = b a  2 0  0 x0 ∈/ (a, b)

The Dirac Delta function 9 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. A bit of history

Sim´eonDenis Poisson (1781-1840)

In 1815 Poisson already for sees the δ(x − x0) as a selector function using for this purpose Lorentzian functions. Cauchy (1823) also made use of selector function in much the same way as Poisson and Fourier gave a series representation on the delta function(More on this later).

Paul Adrien Maurice Dirac (1902-1984)

Dirac ”rediscovered” the delta function that now bears his name in analogy for the continuous case with the Kronecker delta in his seminal works on quantum mechanics.

The Dirac Delta function 10 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. What does it look like ?

p x ∈ (x0 − 1/2p, x0 + 1/2p) δp(x − x0) = 0 x∈ / (x0 − 1/2p, x0 + 1/2p)

The Dirac Delta function 11 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. What does it look like ?

p R x0+1/2p f(x)dx x0−1/2p

The Dirac Delta function 12 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. What does it look like ?

R ∞ R 1/2p −∞ δp(x) dx = −1/2p p dx = 1

Deﬁnition (Dirac delta function)

∞ x = x0 δ(x − x0)dx = 0 x 6= x0

The Dirac Delta function 13 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. What does it look like ?

Deﬁnition (Dirac delta function) ∞ x = x0 δ(x − x0) = 0 x 6= x0

The above expression is just ”formal”, the δ(x) must be always understood in the context of its selector property i.e. within the integral

δ(x) is deﬁned more rigorously in terms of a distribution or a functional (generalized function)

The Dirac Delta function 14 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac delta function as the limit of a family of functions

The Dirac delta function can be pictured as the limit in a sequence of functions δp which must comply with two conditions: R ∞ l´ımp→∞ −∞ δp(x)dx = 1: Normalization condition

δp(x6=0) l´ımp→∞ = 0 Singularity condition. l´ımx→0 δp(x)

The Dirac Delta function 15 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac delta function as the limit of a family of functions

The Dirac delta function can be pictured as the limit in a sequence of functions δp which must comply with two conditions: R ∞ l´ımp→∞ −∞ δp(x)dx = 1: Normalization condition

δp(x6=0) l´ımp→∞ = 0 Singularity condition. l´ımx→0 δp(x)

The Dirac Delta function 15 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Gaussian functions

∆p x H L

p Π "############# δp(x) Gaussian family

r 1 p p 2 δp(x) = exp (−px ) ã Π π &''''''''

x -1 p 1 p "###### "######

The Dirac Delta function 16 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Gaussian functions

Normalization condition

Z ∞ r Z ∞ r Z ∞ p 2 1 2 √ δp(x)dx = exp (−px )dx = exp (−px )d( px) = −∞ π −∞ π −∞ r r 1 Z ∞ 1 Z ∞ exp (−t2)dt = 2 exp (−t2)dt π −∞ π 0

R ∞ −t2 I = 0 e dt 2 R ∞ −y2 R ∞ −z2 RR ∞ 2 2 I = 0 e dy 0 e dz = 0 exp (y + z )dydz

The Dirac Delta function 17 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref.

Normalization condition

 2 2 2   r = y + z  y = r cos φ  z = r sin φ 

2 R π/2 R ∞ −r2 π R ∞ −s π I = 0 dφ 0 e rdr = 4 0 e ds = 4 √ π I = 2

R ∞ −∞ δp(x)dx = 1

The Dirac Delta function 18 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Gaussian functions

Singularity condition

δ (x 6= 0) l´ım p = p→∞ l´ımx→0 δp(x) 2 p p e−px l´ım π = 0 p→∞ p p π

The Dirac Delta function 19 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Lorentzian functions

∆p x H L p Π

δp(x) Lorentzian family p Π 1 p 2 δ (x) = p p 1 + p2x2

x -1 p 1 p

The Dirac Delta function 20 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Lorentzian functions

Normalization condition

Z ∞ 1 Z ∞ pdx 1 Z ∞ dt δp(x)dx = 2 2 = 2 = −∞ π −∞ 1 + p x π −∞ 1 + t

1 t=k 2 l´ım arctan t|t=−k = l´ım arctan k = 1 π k→∞ π k→∞

Π2

ArcTanHxL

The Dirac Delta function 21 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Lorentzian functions

Singularity condition

δ (x 6= 0) l´ım p = p→∞ l´ımx→0 δp(x) p 1 2 2 l´ım 1+p x = π p→∞ p 1 1 l´ım = 0 π p→∞ 1 + p2x2

The Dirac Delta function 22 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Sinc functions

∆pHxL

p Π

δp(x) Sinc family

p sin px Π δp(x) = 2 π px p x Π Π 3 p p

The Dirac Delta function 23 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Sinc functions

Normalization condition

Z ∞ 1 Z ∞ sin (px)dx δp(x)dx = = −∞ π −∞ x 1 Z ∞ sin z dz = π −∞ z 2 Z ∞ sin z 2 π dz = = 1 π 0 z π 2

The Dirac Delta function 24 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. ... as the limit of Sinc functions

Singularity condition

δ (x 6= 0) l´ım p = p→∞ l´ımx→0 δp(x) 1 sin (px) π x sin (px) l´ım p = l´ım = 0 p→∞ p→∞ π px

δp(x) alternative deﬁnition of the Sinc family 1 R p ±itx δp(x) = 2π −p e dt 1 R p δp(x) = π 0 cos (tx)dt

The Dirac Delta function 25 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Properties of the Dirac delta function

Let us denote by xn the roots of the equation f(x) = 0 and suppose that 0 f (xn) 6= 0 then

Composition of functions

P δ(x−xn) δ(f(x)) = 0 n |f (xn|

The Dirac Delta function 26 / 45 2 2 δ(x−a)+δ(x+a) δ(x − a ) = 2|a| .

R ∞ P g(xn) g(x)δ(f(x))dx = n 0 −∞ |f (xn)|

Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref.

Important consequences of the composition property are δ(−x) = δ(x) (symmetry property).

δ(x) δ(ax) = |a| (scaling property).

x0 δ(x− a ) δ(ax − x0) = |a| (a more general formulation of the scaling property).

The Dirac Delta function 27 / 45 R ∞ P g(xn) g(x)δ(f(x))dx = n 0 −∞ |f (xn)|

Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref.

Important consequences of the composition property are δ(−x) = δ(x) (symmetry property).

δ(x) δ(ax) = |a| (scaling property).

x0 δ(x− a ) δ(ax − x0) = |a| (a more general formulation of the scaling property).

2 2 δ(x−a)+δ(x+a) δ(x − a ) = 2|a| .

The Dirac Delta function 27 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref.

Important consequences of the composition property are δ(−x) = δ(x) (symmetry property).

δ(x) δ(ax) = |a| (scaling property).

x0 δ(x− a ) δ(ax − x0) = |a| (a more general formulation of the scaling property).

2 2 δ(x−a)+δ(x+a) δ(x − a ) = 2|a| .

R ∞ P g(xn) g(x)δ(f(x))dx = n 0 −∞ |f (xn)|

The Dirac Delta function 27 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref.

Important consequences of the composition property are δ(−x) = δ(x) (symmetry property).

δ(x) δ(ax) = |a| (scaling property).

x0 δ(x− a ) δ(ax − x0) = |a| (a more general formulation of the scaling property).

2 2 δ(x−a)+δ(x+a) δ(x − a ) = 2|a| .

R ∞ P g(xn) g(x)δ(f(x))dx = n 0 −∞ |f (xn)|

The Dirac Delta function 27 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Convolution

Convolution R ∞ f(x)⊗δ(x+x0) = −∞ f(ς)δ(ς−(x+x0))dς = f(x+x0)

The eﬀect of convolving with the position-shifted Dirac delta is to shift f(t) by the same amount.

R ∞ −∞ δ(ζ − x)δ(x − η)dx = δ(ζ − η))

The Dirac Delta function 28 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Convolution

Convolution R ∞ f(x)⊗δ(x+x0) = −∞ f(ς)δ(ς−(x+x0))dς = f(x+x0)

The eﬀect of convolving with the position-shifted Dirac delta is to shift f(t) by the same amount.

R ∞ −∞ δ(ζ − x)δ(x − η)dx = δ(ζ − η))

The Dirac Delta function 28 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Heaviside

Deﬁnition (Step function) Deﬁnition (Step function) 1 x ≥ 0 Θ(x) = 1 (1 + x ) Θ(x) = 2 |x| 0 x < 0

dΘ 1 d x 1 d 2 = = l´ım arctan (px) = dx 2 dx |x| 2 p→∞ dx π 1 p l´ım = δ(x) π p→∞ 1 + p2x2

Heaviside

dΘ(x) δ(x) = dx

The Dirac Delta function 29 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Heaviside

Deﬁnition (Step function) Deﬁnition (Step function) 1 x ≥ 0 Θ(x) = 1 (1 + x ) Θ(x) = 2 |x| 0 x < 0

dΘ 1 d x 1 d 2 = = l´ım arctan (px) = dx 2 dx |x| 2 p→∞ dx π 1 p l´ım = δ(x) π p→∞ 1 + p2x2

Heaviside

dΘ(x) δ(x) = dx

The Dirac Delta function 29 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Heaviside

y y

p=0.5 x

p=2 x

y y

p=1 x

p=5 x

The Dirac Delta function 30 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Fourier transform of the Dirac delta function

Fourier transform ∗ R ∞ ∗ Γ [δ(x)] = δb(x ) ≡ −∞ δ(x) exp (−2πix x)dx = 1

This property allow us to state yet another deﬁnition of the Dirac delta as the inverse Fourier transform of f(x) = 1

Deﬁnition (Dirac delta function) Z ∞ δ(x) = exp (2πix∗x)dx∗ −∞

The Dirac Delta function 31 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Fourier transform of the Dirac delta function

Fourier transform ∗ R ∞ ∗ Γ [δ(x)] = δb(x ) ≡ −∞ δ(x) exp (−2πix x)dx = 1

This property allow us to state yet another deﬁnition of the Dirac delta as the inverse Fourier transform of f(x) = 1

Deﬁnition (Dirac delta function) Z ∞ δ(x) = exp (2πix∗x)dx∗ −∞

The Dirac Delta function 31 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac delta function obtained from a complete set of orthonormal functions

Let the set of functions {ψn} be a complete system of orthonormal functions in the interval (a, b) and let x and x0 be inner points of that interval. Then

Theorem (Orthonormal functions) P ∗ n ψn(x)ψn(x0) = δ(x − x0)

To proof the theorem we shall demonstrate that the left hand side has the sifting property of the Dirac distribution

R b P ∗ I = a f(x) n ψn(x)ψn(x0)dx = f(x0) P R b ∗ f(x) = m cmψm(x) cm = a f(x)ψm(x)dx

The Dirac Delta function 32 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac delta function obtained from a complete set of orthonormal functions

Z b X X ∗ I = cmψm(x) ψn(x)ψn(x0)dx = a m n Z b X X ∗ cm ψn(x0) ψn(x)ψm(x)dx m n a

R b ∗ K a ψn(x)ψm(x)dx = δmn

X X K I = cm ψn(x0) δmn = n n X cmψm(x0) = f(x0) n

The Dirac Delta function 33 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac comb

Deﬁnition (Dirac comb) P∞ m=−∞ δ(x − ma)

The Dirac Delta function 34 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac comb

The set { √ 1 exp (2πinx/a)} forms a complete set of orthonormal (|a|) functions, then

1 P∞ δ(x) = |a| n=−∞ exp (2πinx/a)

each summand in the LHS in the above expression is periodic with period |a| therefore the whole sum is periodic with the same period and

Dirac comb P∞ 1 P∞ m=−∞ δ(x − ma) = |a| n=−∞ exp (2πinx/a)

The Dirac Delta function 35 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Fourier transform of a Dirac comb

Z ∞ ∞ X ∗ δ(x − ma) exp(−2πix x)dx = −∞ m=−∞ ∞ Z ∞ ∞ X ∗ X ∗ δ(x − ma) exp(−2πix x)dx = exp(2πix ma) m=−∞ −∞ m=−∞

Theorem (Fourier transform of a Dirac comb) P∞ 1 P∞ ∗ ∗ P∞ ∗ ∗ Γ [ m=−∞ δ(x − ma)] = |a| h=−∞ δ(x − h/a) = |a | h=−∞ δ(x − ha )

The Fourier transform of a Dirac comb is a Dirac comb

The Dirac Delta function 36 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Dirac delta in higher dimensional space

Dirac delta in higher dimensions. Cartesian coordinates R R N ··· f(~x)δ(~x − ~x0)d x = f( ~x0)

which comes from

R R ··· f(x1, x1, . . . xN )δ(x1 −x01)δ(x2 −x02) . . . δ(xN −x0N )dx1dx2 . . . dxN = f( 01, x02~x , . . . x0N )

Deﬁnition (Dirac delta in higher dimensions. Cartesian coordinates) QN δ(~x − ~x0) = δ(x1 − x01)δ(x2 − x02) . . . δ(xN − x0N ) = s=1 δ(xs − x0s)

The Dirac Delta function 37 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. General coordinates

{xi}: Cartesian coordinates {yi}: General coordinates

∂x ∂x ∂x x1 = x1(y1, . . . yN ) 1 1 ... 1 ∂y1 ∂y2 ∂yN ∂x2 ∂x2 ∂x2 x2 = x2(y1, . . . yN ) ... ∂y1 ∂y2 ∂yN J(y1, y2, . . . , yN ) = ...... ∂xN ∂xN ... ∂xN xN = xN (y1, . . . yN ) ∂y1 ∂y2 ∂yN

Deﬁnition (Dirac delta in higher dimensions. General coordinates)

1 δ(~x − ~x0) = δ(~y − ~y0) |J(y1,y2,...,yN )|

The Dirac Delta function 38 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Oblique coordinates

x1 = a11y1 + a12y2 . . . a1N yN J(y1, y2, . . . , yN ) =

x2 = a11y1 + a22y2 . . . a2N yN a11 a12 . . . a1N

. a21 a22 . . . a2N ...... xN = aN1y1 + aN2y2 . . . aNN yN aN1 aN2 . . . aNN

  a11 a12 . . . a1N  a21 a22 . . . a2N  ~x =   ~y  ......  aN1 aN2 . . . aNN

Deﬁnition (Dirac delta in higher dimensions. Oblique coordinates)

1 1 δ(~x− ~x0) = δ(~y − ~y0) = √ δ(~y − ~y0) | ||A|| | | ||G|| |

The Dirac Delta function 39 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Recap: Deﬁnitions

d 1 x ∞ x = x0 δ(x) = (1 + ) δ(x − x0)dx = dx 2 |x| 0 x 6= x0

r 1 p p 2 δ(x) = l´ım δ(x) = l´ım exp (−px ) p→∞ p 1 + p2x2 p→∞ π Z ∞ p sin px δ(x) = exp (2πix∗x)dx∗ δ(x) = l´ım −∞ p→∞ π px

X ∗ δ(x − x0) = ψn(x)ψn(x0) n

∗ where {ψn(x)} is a complete set of orthornormal functions.

The Dirac Delta function 40 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Recap: Properties

 1 [f(x−) + f(x+] x ∈ (a, b)  2 0 0 0 Z b  1 + 2 f(x0 ) x0 = a f(x)δ(x − x0)dx = 1 − f(x ) x0 = b a  2 0  0 x0 ∈/ (a, b) Z ∞ X δ(x − xn) X g(xn) δ(f(x)) = 0 g(x)δ(f(x))dx = 0 |f (xn| |f (xn)| n −∞ n x0 δ(x − a ) δ(−x) = δ(x) δ(ax − x0) = |a| δ(x − a) + δ(x + a) δ(x2 − a2) = 2|a| f(x) ⊗ δ(x + x0) = f(x + x0)

∞ ∞ X ∗ X ∗ ∗ ∗ Γ [ δ(x − ma)] = |a | δ(x − ha ) Γ [δ(x)] = δb(x ) = 1 m=−∞ h=−∞ ∞ ∞ X 1 X δ(x − ma) = exp (2πinx/a) |a| m=−∞ n=−∞

The Dirac Delta function 41 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Recap: Higher Dimensions

Cartesian coordinates: Z Z N ··· f(~x)δ(~x − ~x0)d x = f( ~x0)

N Y δ(~x − ~x0) = δ(x1 − x01)δ(x2 − x02) . . . δ(xN − x0N ) = δ(xs − x0s) s=1

General Coordinates {yi}: 1 δ(~x − ~x0) = δ(~y − ~y0) |J(y1, y2, . . . , yN )| Oblique Coordinates ~x = A · ~y: 1 1 δ(~x − ~x0) = δ(~y − ~y0) = δ(~y − ~y0) | ||A|| | p| ||G|| |

The Dirac Delta function 42 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Exercises

Prove:

P δ(x−xn) δ(f(x)) = n 0 (Hint: develop f(x) in Taylor series around xn and |f (xn| prove the sifting property with δ(f(x)))

P∞ ∗ P∞ ∗ n=−∞ f(na) = |x | m=−∞ fb(mx ) (Poisson summation formula)

x−x sin ((2p+1)π 0 ) a P∞ l´ımp→∞ x−x0 = |a| m=−∞ δ(x − x0 − ma) sin (π a )

The Dirac Delta function 43 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. Exercises

Prove that on the limit p → ∞ the sawtooth funtion tends to the Dirac comb

The Dirac Delta function 44 / 45 Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref. References

Prof. RNDr. Jiri Komrska. The Dirac distribution http://physics.fme.vutbr.cz/∼komrska

V. Balakrishnan All about the Dirac Delta Function(?)

The Dirac Delta function 45 / 45