Sinc Function

Author

Eric W. Weisstein
July 5, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SincFunction.nb.

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`
<<Utilities`Typesetting`

Real Line

Plot[Sinc[x],{x,-15,15},
PlotRange->All,
PlotStyle->Red,
AxesLabel->TraditionalForm/@{x,Sinc[x]}]

Complex Plane

ComplexPlot3D[Sinc[z],{z,-5,5}]//Timing

ComplexContourPlot[Sinc[z],{z,-5,5}]//Timing

Coincidence of Cos Maxima

<<Graphics`Colors`

Plot[{Sinc[x],Cos[x]},{x,-15,15},PlotStyle->{Red,Blue},
PlotLabel->TraditionalForm[{
StyleForm[Sinc[x],FontColor->Red],
StyleForm[Cos[x],FontColor->Blue]
}]]

Normalization

Integrate[Sin[x]/x,{x,-∞,∞}]

Integrate[Sin[π x]/(π x),{x,-∞,∞}]

Derivative and Integral

Derivative

D[Sinc[x],x]

Indeterminate and therefore wrong at x==0 :(

Indefinite Integral

Integrate[Sinc[x],x]

Identities

((sin(x))/x)==sinc(x)

FullSimplify[Sin[x]/x]

((sin(n x))/x)==sinc(n x)

FullSimplify[Sin[n x]/(n x)]

((exp(i n x)-exp(-i n x))/((2 i) (n x)))==sinc(n x)

FullSimplify[(1/(n x))((Exp[I n x]-Exp[-I n x])/(2I))]

(Binomial[0, x])==sinc(π x)

FunctionExpand[Binomial[0,x]]

Binomial[0,0]

FunctionExpand[Binomial[0,x]]/.x->0

Extrema

Minimum

Minimize[Sinc[x],x,Reals]

FindMinimum[Sinc[x]&&-10<x<10,x]

FindInstance[Sinc'[x]==0&&0<x<5,x]

FullSimplify[%]

root=Root[{ (Sin[#1]-Cos[#1] #1)&,4.4934094579090641753078920741840006134175044191134722921373`23.598724058603725}];

Sinc[root]//FullSimplify

N[%]

Maximum

Maximize[Sinc[x],x,Reals]

FindMaximum[Sinc[x]&&-10<x<10,x]

Roots

<<Utilities`Plot`

Solve[Sin[x]/x==0,x]

Reduce[Sin[x]/x==0,x]

Reduce[Sinc[x]==0,x]

PlotRoots[Sin[x]/x,{x,0,20},WorkingPrecision->20,PlotStyle->Blue]

Table[n π,{n,6}]//N

Sums

(∑_k^∞)sinc(k)==(1/2) (π-1)

Sum[Sinc[k],{k,∞}]

(∑_k^∞)((sinc(k))^n)

v5.2

Sum[Sin[k]/k,{k,∞}]

FullSimplify[%]

Sum[(Sin[k]/k)^n,{k,∞}]

Timing[s=Table[1/2+Sum[(((((Sin[k])/k)))^n),{k,∞}],{n,10}]//FunctionExpand//FullSimplify//Factor]

Through[{Numerator,Denominator}[Take[s,6]/Pi]]

n=7 case

FullSimplify[FunctionExpand[1/2+Sum[(((((Sin[k])/k)))^7),{k,∞}]]]//Factor

c1=N[%,1000]

c2=1/2+NSum[(((((Sin[k])/k)))^7),{k,∞},PrecisionGoal->1000]

c3=1/2+Sum[(((((Sin[k])/(N[k,1000]))))^7),{k,10000}]

c1-c3

v6

Sum[Sin[k]/k,{k,∞}]

FullSimplify[%]

Sum[(Sin[k]/k)^n,{k,∞}]

Sum[Sinc[k],{k,∞}]

Sum[Sinc[k]^n,{k,∞}]

Timing[s=Table[1/2+Sum[(((((Sin[k])/k)))^n),{k,∞}],{n,10}]//FunctionExpand//FullSimplify//Factor]

Through[{Numerator,Denominator}[Take[s,6]/Pi]]

n=7 case

FullSimplify[FunctionExpand[1/2+Sum[(((((Sin[k])/k)))^7),{k,∞}]]]//Factor

c1=N[%,1000]

c2=1/2+NSum[(((((Sin[k])/k)))^7),{k,∞},PrecisionGoal->1000]

c3=1/2+Sum[(((((Sin[k])/(N[k,1000]))))^7),{k,10000}]

c1-c3

V7

Sum[Sinc[k],{k,∞}]

Sum[((Sinc[k])^2),{k,∞}]

Do[Print[{n,1/2+Sum[Sinc[k]^n,{k,∞}]//Simplify}],{n,10}]

Timing[s=Table[1/2+Sum[Sinc[k]^n,{k,∞}],{n,10}]//FunctionExpand//FullSimplify//Factor]

(∑_(k=-∞)^∞)sinc(k)==1

V7

Sum[Sinc[k],{k,-∞,∞}]

Sum[Sinc[k],{k,-20,20}]//N

V8

Sum[Sinc[k],{k,-∞,∞}]

(∑_(k=-∞)^∞)sinc(π k)==1

V7.0.2

Sum[Sinc[π k],{k,-∞,∞}]

V8

Sum[Sinc[π k],{k,-∞,∞}]

Products

(∏_(k=1)^∞)cos((x/(2^k)))==sinc(x)

Function[x,Product[Cos[x/2^k],{k,∞}]-Sinc[x]]/@{1.,1.3,5.}

v5.2

Product[Cos[x/2^k],{k,∞}]

v6

Product[Cos[x/2^k],{k,∞}]

V7

Product[Cos[x/2^k],{k,∞}]

(∏_(n=1)^∞)(1-((x^2)/((n^2) (π^2))))==sinc(x)

v5.2

Product[1-((x^2)/((n^2)(π^2))),{n,∞}]

FullSimplify[%]

v6

FullSimplify[Product[1-((x^2)/((n^2)(π^2))),{n,∞}]]

v7

Product[1-((x^2)/((n^2)(π^2))),{n,∞}]

(∏_(k=1)^∞)(1-(4/3) (sin^2)((x/(3^k))))==sinc(x)

v5.2

Product[1-(4/3)((Sin[(x/(3^k))])^2),{k,∞}]

v6

Product[1-(4/3)((Sin[(x/(3^k))])^2),{k,∞}]

V7

Product[1-(4/3)((Sin[(x/(3^k))])^2),{k,∞}]

(∏_(k=1)^∞)sinc(((2 π)/(2 k+1)))==((π (Subscript[K, polygon]))/2)

k=N[FromDigits[{8,7,0,0,0,3,6,6,2,5,2,0,8,1,9,4,5,0,3,2,2,2,4,0,9,8,5,9,1,1,3,0,0,4,9,7,1,1,9,3,2,9,7,9,4,9,7,4,2,8,9,2,0,9,2,1,5,9,6,6,7,2,7,8,6,8,3,4,2,9,9,6,4,1,1,4,0,2,5,1,5,9,1,1,8,5,4,4,4,1,4,0,0,9,2,4,9,5,2,8,5,5}]/10^101,102]

((π/2)/k)

NProduct[Sinc[((2π)/(2k+1))],{k,∞}]

v5.2

Product[Sinc[((2π)/(2k+1))],{k,∞}]

NProduct[Sinc[((2π)/(2k+1))],{k,∞}]//Chop

v6

Product[Sinc[((2π)/(2k+1))],{k,∞}]

Product[Sinc[((2π)/(2k+1))],{k,n}]

NProduct[Sinc[((2π)/(2k+1))],{k,∞},WorkingPrecision->20]//Chop

V7

Product[Sinc[((2π)/(2k+1))],{k,∞}]//Timing

Definite Integrals

(∫_(-∞)^∞)((sinc((x/2)))^2) (e^(i x t))dx

Borwein et al. 2004, p. 79

v5

Timing[i1=Integrate[Sinc[x/2]^2Exp[I x t],{x,-∞,∞}]]

FullSimplify[%3,t∈Reals]//Timing

Timing[i2=Integrate[Sinc[x/2]^2Exp[I x t],{x,-∞,∞},Assumptions->t∈Reals]]

Timing[i3=FourierTransform[Sinc[x/2]^2,x,t]]

FullSimplify[i3]

FullSimplify[i3,t∈Reals]

v6

Timing[i1=Integrate[Sinc[x/2]^2Exp[I x t],{x,-∞,∞}]]

FullSimplify[i1,t∈Reals]//Timing

Timing[i2=Integrate[Sinc[x/2]^2Exp[I x t],{x,-∞,∞},Assumptions->t∈Reals]]

Timing[i3=FourierTransform[Sinc[x/2]^2,x,t]]

FullSimplify[i3]

FullSimplify[i3,t∈Reals]

PiecewiseExpand[(Sign[1-t]+t Sign[-1+t]-2 t Sign[t]+(1+t) Sign[1+t]),{},Reals]

(∫_(-∞)^∞)((sin(π x) (e^(i x t)))/(π x))dx

Integrate[((Sin[x])/x) Exp[I x t],{x,-∞,∞}]//Timing

Integrate[((Sin[π x])/(π x)) Exp[I x t],{x,-a,a}]//Timing

FourierTransform[((Sin[x])/x),x,t]

(∫_0^∞)(((((sin(x))/x)))^n)dx

Table[Integrate[(Sin[x]/x)^n,{x,0,∞}],{n,10}]

(π/2)(1+(1/(2^(n-2)))Sum[((((-1))^r)/((r-1)!))((((n-2r))^(n-1))/((n-r)!)),{r,n/2}])

Table[(π/2)(1+(1/(2^(n-2)))Sum[((((-1))^r)/((r-1)!))((((n-2r))^(n-1))/((n-r)!)),{r,n/2}]),{n,10}]

(π/((2^n)(n-1)!))Sum[(((-1))^r)Binomial[n,r](((n-2r))^(n-1)),{r,0,n/2}]

Table[(((2^(-n)) π (∑_(r=0)^(n/2))(((-1))^r) (((n-2 r))^(-1+n)) Binomial[n,r])/((-1+n)!)),{n,10}]

(∫_(-∞)^∞)sinc(π (x-y)) sinc(π y)dy==((sin(π x))/(π x))

Integrate[Sinc[Pi(x-y)]sinc[Pi y],{y,-∞,∞},GenerateConditions->False]

(1/(2 n))(∫_(-n)^n)exp(i t x)dt==sinc(n x)

(1/(2n))Integrate[Exp[I x t],{t,-n,n}]

FullSimplify[%]

%/.x->0

(1/(2n))Integrate[Exp[I x t]/.x->0,{t,-n,n}]

(∫_0^∞)(((sin^a)(x))/(x^b))dx==...

Integrate[(((Sin[x])^a)/(x^b)),{x,0,∞},Assumptions->(a|b)∈Integers&&0<b<a]

(∫_0^∞)((sinc(x))^(2n))dx==((π )/(2 (2 n-1)!))⟨({{2 n-1}, {n-1}})⟩

Integrate[(((Sin[x])^(2n))/(x^(2n))),{x,0,∞}]

Integrate[((Sinc[x])^(2n)),{x,0,∞}]

<<Combinatorica`

Table[(π/(2(2n-1)!))Eulerian[2n-1,n-1],{n,5}]

Table[Integrate[((Sinc[x])^(2n)),{x,0,∞}],{n,5}]