Sine

Author

Eric W. Weisstein
August 23, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Plots

<<Utilities`Plot`

Real Axis

Plot[Sin[x],{x,-2π,2π},
AxesLabel->TraditionalForm/@{x,Sin[x]},
PlotStyle->Red,
Epilog->{Dashing[{.02,.02}],Line[{{-2π,#},{2π,#}}]&/@{-1,1}},
Ticks->{{#,TraditionalForm[#]}&/@{-2π,-π,0,π,2π},{#,TraditionalForm[#]}&/@{-1,-1/2,0,1/2,1}}
]

Re/Im

ComplexPlot3D[Sin[z],{z,-2π,2π}]//Timing

ComplexPlot3D[Sin[z],{z,-2π,2π},
Lighting->False,
ColorFunction->Hue,
PlotPoints->100]

Complex Contour

ComplexContourPlot[Sin[z],{z,-2π,2π}]//Timing

Series

Series[Sin[z],{z,0,10}]

V5

<<DiscreteMath`RSolve`

SeriesTerm[Sin[z],{z,0,n}]

V7

SeriesCoefficient[Sin[z],{z,0,n}]

Table[%,{n,10}]

GeneratingFunction[(Piecewise[{{((I (I^n) (-1+(((-1))^n)))/(2 n!)), n≥0}, {0, True}}]),n,x]

FullSimplify[%]

GeneratingFunction[(((((-1))^(n-1)) )/((2 n-1)!)),n,x]

Derivatives and Integrals

Derivative

D[Sin[x],x]

Integral

Integrate[Sin[x],x]

Sums

(∑_n^∞)(((((-1))^(n-1)) (x^(2 n-1)))/((2 n-1)!))==sin(x)

Sum[((((-1))^(n-1))/((2n-1)!))(x^(2n-1)),{n,∞}]

(∑_(n=0)^∞)(((i (i^n) ((((-1))^n)-1)) (z^n))/(2 n!))==sin(z)

Sum[((I (I^n) (-1+(((-1))^n)))/(2 n!))(z^n),{n,0,∞}]

FullSimplify[%]

(∑_k^(n-1))sin(((π k)/n))==cot((π/(2 n)))

Drane

V6

Sum[Sin[((k π)/n)],{k,n-1}]

V7

Sum[Sin[((k π)/n)],{k,n-1}]

FullSimplify[%]

(∑_(n=0)^N)sin(n x)==csc((x/2)) sin(((N x)/2)) sin((1/2) (N+1) x)

v4.2

FullSimplify[Sum[Sin[n x],{n,0,N}]]

FullSimplify[ExpToTrig[Sum[Sin[k x]^2,{k,0,N}]]]

v5.0

FullSimplify[ComplexExpand[Im[Sum[Exp[I n x],{n,0,N}]]]]

FullSimplify[Sum[Sin[n x],{n,0,N}]]

s=Sum[Exp[I n x],{n,0,N}]

(∑_(k=0)^N)(sin^2)(k x)==(N/2)-(1/2) csc(x) sin(N x) cos((N+1) x)

Sum[((Sin[k x])^2),{k,0,N}]

(∑_(k=0)^n)(Binomial[n, k]) sin((1/2) π (n-k)) (cos^k)(x) (sin^(n-k))(x)==sin(n x)

V6

Sum[Binomial[n,k]Cos[x]^k Sin[x]^(n-k)Sin[(n-k) π/2],{k,0,n}]

FullSimplify[%,n∈Integers]

V7

Sum[Binomial[n,k]Cos[x]^k Sin[x]^(n-k)Sin[(n-k) π/2],{k,0,n}]

FullSimplify[%,n∈Integers]

(∑_(k=0)^∞)((sin(a (2 k+1)))/(((2 k+1))^n))==(((((-1))^((n-1)/2)) (π^n) (Subscript[E, n-1])((a/π)))/(4 (n-1)!))

Sum[((Sin[(2k+1)a])/((2k+1)^n)),{k,0,∞}]//InputForm

Sum[Sin[a*(1 + 2*k)]/(1 + 2*k)^n, {k, 0, Infinity},Assumptions->Mod[n,2]==1]//InputForm

Subtract@@@Table[{((-I)*2^(-1-n)*(-LerchPhi[E^((-2*I)*a),n,1/2]+E^((2*I)*a)*LerchPhi[E^((2*I)*a),n,1/2]))/E^(I*a),((((-1))^((n-1)/2))/(4(n-1)!))(π^n)EulerE[n-1,a/Pi]}/.{a->RandomComplex[],n->(1+2RandomInteger[{1,10}])},{100}]//Chop//Union

Table[{Sum[((Sin[(2k+1)a])/((2k+1)^n)),{k,0,∞}],((((-1))^((n-1)/2))/(4(n-1)!))(π^n)EulerE[n-1,a/Pi]},{n,1,5,2}]//FullSimplify//Column

FunctionExpand[%192]//FullSimplify

%176/.a->1+1.2I//N//Chop

n==1

FullSimplify[(1/2) I (ArcTanh[(E^(-I a))]-ArcTanh[(E^(I a))])]

FullSimplify[(1/2) I (ArcTanh[(E^(-I a))]-ArcTanh[(E^(I a))])-π/4]

Subtract@@@Table[{Pi/4,(1/2) I (ArcTanh[(E^(-I a))]-ArcTanh[(E^(I a))])}/.a->RandomComplex[],{100}]//Chop//Union

n == 3

FullSimplify[(1/16) I (E^(-I a)) (LerchPhi[(E^(-2 I a)),3,(1/2)]-(E^(2 I a)) LerchPhi[(E^(2 I a)),3,(1/2)])]

Subtract@@@Table[{(1/16) I (E^(-I a)) (LerchPhi[(E^(-2 I a)),3,(1/2)]-(E^(2 I a)) LerchPhi[(E^(2 I a)),3,(1/2)]),(1/8) a π (-a+π)}/.a->RandomComplex[],{100}]//Chop//Union

n == 5

FullSimplify[(1/64) I (E^(-I a)) (LerchPhi[(E^(-2 I a)),5,(1/2)]-(E^(2 I a)) LerchPhi[(E^(2 I a)),5,(1/2)])]

Subtract@@@Table[{(1/64) I (E^(-I a)) (LerchPhi[(E^(-2 I a)),5,(1/2)]-(E^(2 I a)) LerchPhi[(E^(2 I a)),5,(1/2)]),(1/96) a π ((a^3)-2 (a^2) π+(π^3))}/.a->RandomComplex[],{100}]//Chop//Union

Limits

-π ((lim_(n->∞)) (1/(ln(n)))(∑_k^∞)((μ(k))/k) frac(((k z)/(2 π))) ln((n/k)))==sin(z)

Michael's Sum

V6

sin[n_,z_]:=-Pi Sum[MoebiusMu[k] Log[n/k]/Log[n]*FractionalPart[k z/(2Pi)]/k,{k,n}]

sin[z_]:=-Pi NSum[MoebiusMu[k] Log[n/k]/Log[n]*FractionalPart[k z/(2Pi)]/k,{k,Infinity},Method->Fit]

sin[n,z]//TraditionalForm

{Sin[#],sin[20000,#]}&[.02]

sin[2,.02]

SequenceLimit[Accumulate[Table[sin[n,.02],{n,1000}]]]

sin[.02]

V7

-Pi Sum[MoebiusMu[k] Log[n/k]/Log[n]*FractionalPart[k z/(2Pi)]/k,{k,n}]//InputForm

Table[{-Pi/Log[n] Sum[MoebiusMu[k]/k Log[n/k]FractionalPart[k z/(2Pi)],{k,n}],Sin[z]}/.{n->10^6,z->RandomReal[]},{2}]

Products

π s (∏_(n=1)^∞)(1-((s^2)/(n^2)))==sin(π s)

V6

π s Product[1-((s^2)/(n^2)),{n,∞}]

Simplify[%,s∈Reals]

V7

π s Product[1-((s^2)/(n^2)),{n,∞}]

x (∏_n^∞)(1-((x^2)/((π^2) (n^2))))==sin(x)

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

FullSimplify[%]

(∏_(k=1)^(n-1))sin(((k π)/n))==(2^(1-n)) n

V6

Product[Sin[((k π)/n)],{k,n-1}]

V7

Product[Sin[((k π)/n)],{k,n-1}]

Sum-Product Identities

(∑_j^n)(∏_k^n)If[k==j,1,((sin(θ(j)-θ(k)+α))/(sin(θ(j)-θ(k))))]==((sin(n α))/(sin(α)))

Calogero's Identity

V6

SinIdentity[n_,{φ_,θ_,α_}]:=Sin[n α]/Sin[α]-Sum[Product[If[k==j,1,Sin[θ[j]-θ[k]+α]/Sin[θ[j]-θ[k]]],{k,n}],{j,n}]

Simplify[SinIdentity[3,{ϕ,θ,α}]]

V7

Sum[Product[If[k==j,1,Sin[θ[j]-θ[k]+α]/Sin[θ[j]-θ[k]]],{k,n}],{j,n}]//Timing

Product[If[k==j,1,Sin[θ[j]-θ[k]+α]/Sin[θ[j]-θ[k]]],{k,n}]

Integrals

(∫_0^∞)sin((x^n))dx==Γ((1/n)+1) sin((π/(2 n)))

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

FullSimplify[Gamma[1+(1/n)] Sin[(π/(2 n))]-((1/n))! Sin[(π/(2 n))]]

Continued Fractions

(x/(((x^2)/(ContinuedFractionK[(2 n) (2 n+1) (x^2),(2 n+2) (2 n+3)-(x^2),{n,∞}]+(2 3-(x^2))))+1))==sin(x)

x/(1+x^2/((2 3-x^2)+ContinuedFractionK[(2n)(2n+1)x^2,(2n+2)(2n+3)-x^2,{n,∞}]))

x/(1+x^2/((2 3-x^2)+ContinuedFractionK[(2n)(2n+1)x^2,(2n+2)(2n+3)-x^2,{n,10}]))/.x->.4

Sin[.4]