q-Cosine

Author

Eric W. Weisstein
October 2, 2008

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

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

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

Code

Sums

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

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

Qcos[z,q]

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

Sum[(((((-1))^n)(z^(2n)))/(QPochhammer[q,q,2n])),{n,0,20}]/.{q->.2,z->.3}

(Subscript[(SuperStar[cos]), q])(π z)==(((q^(z^2)) (QPochhammer2[(q^(1-2 z)), (q^2)]) (QPochhammer2[(q^(2 z+1)), (q^2)]))/((((QPochhammer2[q, (q^2)])))^2))==((EllipticTheta[2, z, exp(((π^2)/(ln(q))))])/(EllipticTheta[2, 0, exp(((π^2)/(ln(q))))]))

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

With[{z=.4},((((q^(z^2)) QPochhammer[(q^(1-2 z)),(q^2)] QPochhammer[(q^(1+2 z)),(q^2)])/((QPochhammer[q,(q^2)])^2))/.z->(1/2-z))/.q->.9999]

Cos[.4 Pi]

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

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

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

((EllipticTheta[2, π z, exp(((π^2)/(ln(q))))])/(EllipticTheta[2, 0, exp(((π^2)/(ln(q))))]))

Gosper

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

With[{z=.1},((EllipticTheta[2,z,Exp[((π^2)/(Log[q]))]])/(EllipticTheta[2,0,Exp[((π^2)/(Log[q]))]]))/.q->.99]

Cos[.1]