Integrating QPochhammer[q,q] over the unit disk (|q| <1) gave me 0, after dodecasecting the pentagonal number series. Integrating q^(1/24) QPochhammer[q,q] instead (Dedekind eta in q form) got me (after FullSimplify heroics) π √(1 - √(2 + √3)/2) (√3 π (-1 + 2 Cosh[4 π/√3]) - Sinh[√3 π])/(2 Cosh[4 π/√3] - 1) ~ 3.108230164891045443, presumably π times the average of Dedekind eta over the disk. Anybody feel like checking? The natural boundary seems to preclude any convincing numerics. (Again defeating ries at -l7, which only got to 2⨉10^-14.) —rwg On Tue, Jul 2, 2019 at 5:47 PM Bill Gosper <billgosper@gmail.com> wrote:
Surprise: Integrate[DedekindEta[-((I*Log[q])/(2*Pi))], {q, 0, 1}] == (2 √2 π Sinh[Sqrt[2/3] π])/(-1 + 2 Cosh[2 Sqrt[2/3] π])
Surprisingly easy derivation: CloudObject[" https://www.wolframcloud.com/obj/3ed3a0a9-14f1-42e1-8c6a-6f7fd0bbf483"]
On Sun, Jun 23, 2019 at 1:12 AM Bill Gosper <billgosper@gmail.com> wrote:
Out[10]= {-(Log[7]/(2 ProductLog[-1, -(((5/(1 + E^3 Sqrt[\[Pi]]))^(1/4) Log[7])/( 2 \[Pi]))])), (8 Sqrt[3/23] \[Pi] Sinh[(Sqrt[23] \[Pi])/6])/(-1 + 2 Cosh[(Sqrt[23] \[Pi])/3])}
In[11]:= N@%
Out[11]= {0.368412535931434, 0.368412535931434}
In[12]:= Equal @@ %%
Out[12]= False The 2nd expression is, amazingly, Integrate[QPochhammer[q,q],{q,0,1}]. https://en.wikipedia.org/wiki/Euler_function I can't quite coax it out of ries.
Does anyone have a closed form for Integrate[DedekindEta[Log[q]/2/I/π],{q,0,1}] ~ .3416968346182687559823? It will probably involve Gamma (which is not yet in ries).
No Gamma. But ries still falls short. Depth setting -l7 can't hack both sinh and cosh. -l8 runs me out of storage. Probably not if ries optimized the common subexpression √(2/3) π
—rwg