Shah Function

Author

Eric W. Weisstein
March 26, 2010

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

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

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

Notation

DiracComb[x]//TraditionalForm

Values

Table[DiracComb[x],{x,-5/2,5/2,1/2}]

Table[DiracDelta[x],{x,-5,5}]

Plot

MakeBoxes[DiracComb[x_],TraditionalForm]:=RowBox[{"∐",AdjustmentBox["∐",BoxMargins->{{-0.6,0.6},{0.,0.}}],AdjustmentBox[RowBox[{"(",MakeBoxes[x,TraditionalForm],")"}],BoxMargins->{{-.9,.9},{0.,0.}}]}]

DiracComb[x]//TraditionalForm

With[{n=3,eps=.4},
Plot[0,{x,-5,5},PlotRange->{{-#,#}&[n+eps],{-.1,1.1}},
AxesLabel->{x,DiracComb[x]},
Epilog->{
Red,Line[{{-#,0},{#,0}}]&[n+eps],
Arrow[{{#,0},{#,1}}]&/@Range[-n,n]
}
]
]

G.F. Identities

∐∐(x)->δ(x)

FullSimplify[DiracComb[x],-1/2<x<1/2]

V8

FullSimplify[DiracComb[x],-1/2<x<1/2]

∐∐(-x)==∐∐(x)

DiracComb[-x]

∐∐(n+x)==∐∐(x)

Table[DiracComb[x+n],{n,-5,5}]

FullSimplify[DiracComb[x+n],n∈Integers]

∐∐(2 x)==??

DiracComb[2x]

FunctionExpand[%]

DiracDelta[2x]

∐∐(x-(1/2))==∐∐(x+(1/2))

DiracComb[x-1/2]

DiracComb[x+1/2]

FullSimplify[DiracComb[x-1/2]-DiracComb[x+1/2]]

FullSimplify[DiracComb[x-1/2]-DiracComb[x+1/2],x∈Reals]

FullSimplify[DiracComb[x-1/2]-DiracComb[x+1/2],x∈Integers]

∐∐(x,y)==∐∐(x) (∐∐(y))

FunctionExpand[DiracComb[x,y]]

Identities

∐∐(x)==π (DiracDeltaSeq[sin(π x)])

FullSimplify[π DiracDelta[Sin[π x]]]

Limit[DiracDelta[x],x->0]

DiracDelta[x]/.x->0

DiscretePlot[Limit[π DiracDelta[Sin[π y]],y->x],{x,-3,3,1/10}]

Integrals

(∫_(n-1/2)^(n+1/2))∐∐(x)dx==1

Integrate[DiracDelta[x],{x,-1/2,1/2}]

Integrate[DiracComb[x],{x,-1/2,1/2}]

Integrate[DiracComb[x],{x,-1/2,1/2}]

V8

Integrate[DiracDelta[x],{x,-1/2,1/2}]

Integrate[DiracComb[x],{x,-1/2,1/2}]

Integrate[DiracComb[x],{x,-1/2,1/2}]

Single Sums

∐∐(x)=(∑_(n=-∞)^∞)δ(x-n)

v4.2

Shah[x_]:=Sum[DiracDelta[x-n],{n,-∞,∞}]

Integrate[Shah[x],{x,n-1/2,n+1/2}]

Integrate[Sum[DiracDelta[x-n],{n,-∞,∞}],{x,n-1/2,n+1/2}]

Integrate[DiracDelta[x-n],{x,n-1/2,n+1/2}]

Plot[-UnitStep[-(1/2)-n]+UnitStep[(1/2)-n],{n,-1,1}]

Sum[Integrate[DiracDelta[x-n],{x,-1/2,1/2}],{n,-∞,∞}]

v5.0

$Version

Shah[x_]:=Sum[DiracDelta[x-n],{n,-∞,∞}]

Integrate[Shah[x],{x,-1/2,1/2}]

Integrate[DiracDelta[x-n],{x,-1/2,1/2}]

Plot[UnitStep[(1/2)-n] UnitStep[(1/2)+n],{n,-1,1}]

Sum[UnitStep[1/2-n] UnitStep[1/2+n],{n,-∞,∞}]//InputForm

V7

Sum[DiracDelta[x-n],{n,-∞,∞}]

(∑_(n=-∞)^∞)f(n) DiracComb(x-n)==DiracComb(x) f(x)

Sum[f[n]DiracComb[x-n],{n,-∞,∞}]

V8

Sum[f[n]DiracComb[x-n],{n,-∞,∞}]

(1/(Abs[a]))(∑_(n=-∞)^∞)(DiracDeltaSeq[x-(n/a)])==∐∐(a x)

(1/(Abs[a]))Sum[DiracDelta[x-(n/a)],{n,-∞,∞}]

V8

(1/(Abs[a]))Sum[DiracDelta[x-(n/a)],{n,-∞,∞}]

(∑_(k=-∞)^∞)exp(2 π i k x)==∐∐(x) [wrong]

From MFID; but wrong

Sum[Exp[2π I x k],{k,-∞,∞}]

FullSimplify[Sum[Exp[2π I x k],{k,-n,n}]]

Limit[Table[Csc[π x] Sin[(1+2 n) π x],{n,20}],x->0]

Table[Csc[π x] Sin[(1+2 n) π x]/.x->3/2,{n,5}]

Table[Csc[π x] Sin[(1+2 n) π x]/.x->3/16,{n,5}]//N

Plot[Table[Csc[π x] Sin[(1+2 n) π x],{n,5}],{x,-3,3},PlotStyle->{Red,Yellow,Green,Blue,Purple},Evaluated->True]

1+2(∑_(k=-∞)^∞)cos(2 π k x)==∐∐(x) [wrong]

Sum[1+2Cos[2π k x],{k,-∞,∞}]

FullSimplify[Sum[1+2Cos[2π k x],{k,-n,n}]]

Plot[Table[1+2 n+2 Csc[π x] Sin[(1+2 n) π x],{n,5}],{x,-3,3},PlotStyle->{Red,Yellow,Green,Blue,Purple},Evaluated->True]

Double Sums

(∑_(n=-∞)^∞)(∑_(m=-∞)^∞)DiracComb(x-m,y-n)==DiracComb(x,y)

Sum[DiracComb[x-m,y-n],{n,-∞,∞},{m,-∞,∞}]

V8

Sum[DiracComb[x-m,y-n],{n,-∞,∞},{m,-∞,∞}]

Limits

(lim_(ϵ->0)) (EllipticTheta[3, -π x, (e^(-4 (π^2) ϵ))])==W(x)

Plot[EllipticTheta[3,-π x,Exp[-4(π^2)#]]&/@{.0001,.0005,.001,.01,.1},{x,-2,2},PlotRange->All,PlotStyle->{Red,Yellow,Green,Blue,Purple},Evaluated->True]

(lim_(ϵ->0)) (EllipticTheta[3, -π x, (e^(-2 (π^2)i ϵ))])==W(x)

Plot[Re[EllipticTheta[3,-π x,Exp[-2(π^2)I#]]]&/@{.1},{x,-2,2},PlotRange->All,PlotStyle->{Red,Yellow,Green,Blue,Purple},Evaluated->True]

-2 π^2 I .1

TimeConstrained[EllipticTheta[3,2π ,Exp[-2(π^2)I.1]],300]//Timing

(lim_(ϵ->0)) (1/(π ϵ))(∑_(k=-∞)^∞)sin(((x-k)/ϵ))==W(x)

From MFID, but apparently wrong

Sum[Sin[((x-k)/ϵ)],{k,-∞,∞}]

FullSimplify[Sum[Sin[((x-k)/ϵ)],{k,-n,n}]]

Plot[Evaluate[With[{n=100,ϵ=.001},(1/(π ϵ))Csc[(1/(2 ϵ))] Sin[((1+2 n)/(2 ϵ))] Sin[(x/ϵ)]]],{x,-2,2}]