q-Sine

Author

Eric W. Weisstein
October 2, 2008

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

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

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

Code

Sums

(Subscript[sin, q])(z)==(∑_(n=0)^∞)(((((-1))^n) (z^(2 n+1)))/(QPochhammer[q, q, 2 n+1]))

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

Sum[(((((-1))^n)(z^(2n+1)))/(QPochhammer[q,q,2n+1])),{n,0,30}]/.{q->.2,z->.3}

Qsin[z,q]/.{q->.2,z->.3}

Qsin[z,q]

Limit[Qsin[z,q],q->1]

expr=Qsin[z,q]/.QHypergeometricPFQ[{0},{},qq_,zz_]:>1/QPochhammer[zz,qq]

N[expr/.{q->1-10^-4,z->3/10}]

Sin[.3]

(Subscript[(SuperStar[sin]), q])(π z)==(((q^(((z-(1/2)))^2)) (QPochhammer2[(q^(2 z)), (q^2)]) (QPochhammer2[(q^(2-2 z)), (q^2)]))/((((QPochhammer2[q, (q^2)])))^2))==((EllipticTheta[1, π z, p])/(EllipticTheta[1, (π/2), p]))

(q^(((z-1/2))^2))Product[(((1-(q^(2n-2z)))(1-(q^(2n+2z-2))))/(((1-(q^(2n-1))))^2)),{n,∞}]

-(((((-1+q))^2) (q^((((-(1/2)+z))^2)+2 z)) QPochhammer[(q^(-2 z)),(q^2)] QPochhammer[(q^(-2+2 z)),(q^2)])/((-1+(q^(2 z))) (-(q^2)+(q^(2 z))) ((QPochhammer[(1/q),(q^2)])^2)))/.{q->.2,z->.3/π}

(((q^(((z-1/2))^2))QPochhammer[(q^(2z)),(q^2)]QPochhammer[(q^(2-2z)),(q^2)])/((QPochhammer[q,(q^2)])^2))/.{q->.2,z->.3/π}

Limit[(((q^(((z-1/2))^2))QPochhammer[(q^(2z)),(q^2)]QPochhammer[(q^(2-2z)),(q^2)])/((QPochhammer[q,(q^2)])^2)),q->1]

With[{z=.1},(((q^(((z-1/2))^2))QPochhammer[(q^(2z)),(q^2)]QPochhammer[(q^(2-2z)),(q^2)])/((QPochhammer[q,(q^2)])^2))/.q->.9999]

Sin[.1 Pi]

i (q^(z^2))(( (EllipticTheta[1, i z ln(q), q]))/(EllipticTheta[4, 0, q]))

I (q^(z^2))((EllipticTheta[1,I z Log[q],q])/(EllipticTheta[4,0,q]))/.{q->.2,z->.3/π}

((EllipticTheta[1, π z, exp(((π^2)/(ln(q))))])/(EllipticTheta[1, (π/2), exp(((π^2)/(ln(q))))]))

Solve[Log[p]Log[q]==(π^2),p]

Gosper

((EllipticTheta[1,π z,Exp[((π^2)/(Log[q]))]])/(EllipticTheta[1,(1/2)π,Exp[((π^2)/(Log[q]))]]))/.{q->.2,z->.3/π}

PI[q_]:=((pi[q])/(1-(q^2)))

((PI[q])/(PI[(q^2)]))//Simplify