Delta Function

Author

Eric W. Weisstein
March 26, 2010

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GeneralizedFunctions/DeltaFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DeltaFunction.html.

©2010 Wolfram Research, Inc. except for portions noted otherwise

H'(x) Definition

v5.2

D[UnitStep[x],x]

v6

D[UnitStep[x],x]

D[HeavisideTheta[x],x]

Example

DiracDelta[x^2+x-2]//FunctionExpand

Identities

FunctionExpand[DiracDelta[x,y]]

Integrals

Fundamental Integral

v5

Integrate[f[x]DiracDelta[x-a],{x,-∞,∞}]

v6

Integrate[f[x]DiracDelta[x-a],{x,-∞,∞},Assumptions->a∈Reals]

Epsilon

i=Integrate[f[x]DiracDelta[x-a],{x,a-ϵ,a+ϵ}]

Simplify[i,#]&/@{ϵ<0,ϵ==0,ϵ>0}

Simplify[i,ϵ∈Reals]

Integrate[f[x]DiracDelta[x-a],{x,a-ϵ,a+ϵ},Assumptions->ϵ>0]

Integrate[f[x]DiracDelta[x-a],{x,a-ϵ,a+ϵ},Assumptions->ϵ∈Reals]

Degenerate Cases

v4.2.1

Integrate[DiracDelta[x],{x,Sequence@@#}]&/@{{-∞,∞},{-∞,0},{0,∞}}

v5

Integrate[DiracDelta[x],{x,Sequence@@#}]&/@{{-∞,∞},{-∞,0},{0,∞}}

v6

Integrate[DiracDelta[x],{x,Sequence@@#}]&/@{{-∞,∞},{-∞,0},{0,∞}}

Multidimensional

Integrate[DiracDelta[x,y],{x,-∞,∞},{y,-∞,∞}]

Integrate[DiracDelta[x,y,z],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]

(∫_(-1)^1)δ((1/z))dz

4.1

Integrate[DiracDelta[1/z],{z,-1,1}]

Integrate[DiracDelta[1/z],{z,-∞,∞}]

5.0

Integrate[DiracDelta[1/z],{z,-1,1}]

Integrate[DiracDelta[1/z],{z,-∞,∞}]

v6

Integrate[DiracDelta[1/z],{z,-1,1}]

Integrate[DiracDelta[1/z],{z,-ϵ,ϵ}]

Derivatives

Integrate[f[x]DiracDelta'[x-a],{x,-∞,∞}]

Simplify[DiracDelta'[-x]+DiracDelta'[x]]

Simplify[x^2 DiracDelta'[x]]

FullSimplify[DiracDelta[x-a],(a|x)∈Reals&&a≠x]

FullSimplify[DiracDelta[a x]-((DiracDelta[x])/(Abs[a])),(a|x)∈Reals&&a≠0]

Integrals

Integrate[DiracDelta[x^2-a^2],{x,-∞,∞}]

Integrate[(1/(2Abs[a]))(DiracDelta[x+a]+DiracDelta[x-a]),{x,-∞,∞}]

i1=Integrate[(1/(2Abs[a]))(DiracDelta[x+a]+DiracDelta[x-a]),{x,x1,x2}]

i2=Integrate[DiracDelta[x^2-a^2],{x,x1,x2 }]

ODE

DSolve[(ode=-6 y[x]+(4-6 x) y'[x]+(1-x) x y''[x])==0,y,x]

dsolGD=Function[{x},C DiracDelta''[x]]

ode/.{y->dsolGD}

FullSimplify[%]

Fourier Series

Sum[Cos[n(x-a)],{n,k}]

Sum[Cos[n(x-a)],{n,∞}]

Plot[1/(2π)+Sum[Cos[n(x-1)],{n,10000}]/Pi,{x,-10,10},PlotRange->{-1,1},
PlotStyle->Red]

Fourier Transform

FourierTransform[1,x,k,FourierParameters->{0,-2π}]

FourierTransform[DiracDelta[x],x,k,FourierParameters->{0,-2π}]

FourierTransform[DiracDelta[x-x0],x,k,FourierParameters->{0,-2π}]

InverseFourierTransform[DiracDelta[x],x,k,FourierParameters->{0,-2π}]

Multidimensional

Integrate[DiracDelta[x,y],{x,-∞,∞},{y,-∞,∞}]

Integrate[DiracDelta[x,y,z],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]

Limits

(f0={
1/Pi ϵ/(x^2+ϵ^2),
(1/2)ϵ Abs[x]^(ϵ-1),
1/2/Sqrt[Pi ϵ] Exp[-x^2/4/ϵ],
1/Pi/x Sin[x/ϵ],
1/ϵ AiryAi[x/ϵ],
1/ϵ Re[BesselJ[1/ϵ,(x+1)/ϵ]],Abs[1/ϵ Exp[-x^2/ϵ]LaguerreL[n,2x/ϵ]]
}
)//ColumnForm//TraditionalForm

Print[{#,Limit[Limit[Integrate[#,{x,-L,L},Assumptions->L>0&&ϵ>0],ϵ->0,Assumptions->L>0],L->0]//Timing}]&/@f0

V5

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
Partition[Plot[Evaluate[Table[#/.{ϵ->e,n->1},{e,.1,.2,.02}]],{x,-3,3},PlotRange->If[#===ϵ Abs[x]^(ϵ-1),{Automatic,{0,3}},{Automatic,All}],
AxesLabel->TraditionalForm/@{x,HoldForm[#]/.n->1},
PlotStyle->{Red,Orange,Yellow,Green,Blue,Violet}]&/@f0,3,3,{1,1},{}]]],
GraphicsSpacing->{-.15,.2}
]

V6

<<Graphics`Colors`

GraphicsGrid[
Partition[Plot[Evaluate[Table[#/.{ϵ->e,n->1},{e,.1,.2,.02}]],{x,-3,3},PlotRange->If[#===ϵ Abs[x]^(ϵ-1)/2,{Automatic,{0,1}},{Automatic,All}],
AxesLabel->{x,HoldForm[#]/.n->1},
PlotStyle->{Red,Orange,Yellow,Green,Blue,Violet}]&/@f0,3,3,{1,1},{}],Spacings->{-70,Automatic},ImageSize->600]

Limits 2

Limit[f0,ϵ->0]//ColumnForm

Limit[f0,ϵ->0,Assumptions->x!=0]//ColumnForm

Limit[f0,ϵ->0,Assumptions->x∈Reals]//ColumnForm

finf={1/2/Pi Sin[(n+1/2)x]/Sin[x/2]};

Limit[finf,n->∞]

Limit[finf,n->∞,Assumptions->x∈Reals]

<<Graphics`Colors`

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
Partition[Plot[Evaluate[Table[#/.{n->nn},{nn,10,50,10}]],{x,-1,1},PlotRange->All,
AxesLabel->TraditionalForm/@{x,#},
PlotStyle->{Red,Orange,Yellow,Green,Blue,Violet}]&/@finf,3,3,{1,1},{}]]],
GraphicsSpacing->{-.15,.2}
]

With[{n=500},Plot[1/2/Pi Sin[(n+1/2)x]/Sin[x/2],{x,-.2,.2},PlotRange->{-20,20}]]