prompts me to t(rot )out these four old 𝜗 half-angle formulas: Out[419]= {EllipticTheta[1, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[2, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] - EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[3, x/2, q] == (1/(2^( 1/4)))((EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4)), EllipticTheta[4, x/2, q] == (1/(2^( 1/4)))((-EllipticTheta[2, 0, q]^3 EllipticTheta[2, x, q] + EllipticTheta[3, 0, q]^3 EllipticTheta[3, x, q] + EllipticTheta[4, 0, q]^3 EllipticTheta[4, x, q])^(1/4))} Check: In[432]:= Assuming[0 < x < \[Pi]/2 && 0 < q < 1, FullSimplify@Series[%419, {q, 0, 22}, {x, 0, 22}]] // tim During evaluation of In[432]:= 419.850403,4 (*Seven minutes*). Out[432]= {True, True, True, True} —rwg