To summarize, expressing eta in terms of theta, EllipticTheta[1, \[Pi]/3, q] == EllipticTheta[2, \[Pi]/6, q] == Sqrt[3] q^(1/4) EllipticTheta[4, 3/2 I Log[q], q^9] == Sqrt[3] DedekindEta[-((3 I Log[q])/\[Pi])] (long known but often forgotten). Switching theta[1] with theta[2] (or pi/3 with pi/6) gives you *five* etas instead of one: EllipticTheta[1, \[Pi]/6, q] == EllipticTheta[2, \[Pi]/3, q] == ( DedekindEta[-((I Log[q])/\[Pi])]^2 DedekindEta[-((6 I Log[q])/\[Pi])])/( DedekindEta[-((2 I Log[q])/\[Pi])] DedekindEta[-((3 I Log[q])/\[Pi])]) --rwg On Fri, Jan 1, 2016 at 8:18 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Fri, Dec 25, 2015 at 4:55 AM, rwg <rwg@sdf.org> wrote:
On 2015-12-25 04:13, Bill Gosper wrote:
https://en.wikipedia.org/wiki/Dedekind_eta_function gives a peculiar theta expression for Dedekind eta.
Even more peculiar for not being range-reduced: theta3[π(τ+1)/2] instead of theta4[πτ/2]. --rwg
Combining with the more obvious
DedekindEta[\[Tau]] -> EllipticTheta[1, Pi/3, E^((I Pi \[Tau])/3)]/Sqrt[3] gives the weird-looking EllipticTheta[1,Pi/3, q] == -Sqrt[3] q^(25/4) EllipticTheta[4, 15/2 I Log[q], q^9] == (Sqrt[Pi/3] EllipticTheta[1, Pi/3, E^(Pi^2/(9 Log[q]))])/Sqrt[-Log[q]] (The latter via Jacobi's imaginary transformation.) --rwg
The latter is equivalent to the usual modular relation DedekindEta[-1/\[Tau]] == Sqrt[-I \[Tau]] DedekindEta[\[Tau]] . --rwg