I want to ascribe the expression below to RWG. How should I cite a math-fun mail? Using "priv.comm." now, but that is not really true. I neither like "unpublished note". Any idea? * rwg@sdf.lonestar.org <rwg@sdf.lonestar.org> [Feb 26. 2009 10:36]:
[...]
Enormous screwing around gave
5 %i %pi - -------- 2 8 2 2 48 4 4 eta (q ) %i eta (q ) 1/4 eta(%i q) = %e eta(q ) (---------- + -----------) , 2 2 2 8 eta (q ) eta (q )
[...]
I restate in \Eta(x) = \prod_{n=1}{\infty}{1-x^n} and give two more expressions: 1) RWG gives: # 4 [ 1 ]1/4 # \Eta(i x)= \Eta(x ) | - - 4 i x u | # [ u ] # # 8 2 # \Eta(x ) # \where u = --------- # 2 2 # \Eta(x ) # 2) Set P to # 4 8 # \Eta(x ) # \Eta(+i x) \Eta(-i x) = --------------------- # 2 3 8 3 # \Eta(x ) \Eta(x ) # and set R to # 2 2 8 7 # \Eta(+i x) \Eta(x ) \Eta(x ) # ---------- = ---------------------- # \Eta(-i x) 4 7 16 2 # \Eta(x ) \Eta(x ) # then # +---+ [ +---+ +---+ ] # \Eta(i x)=\|P/2 [ \|1+R - i \|1-R ] # 3) By solving # 16 4 8 8 4 16 2 24 # \Eta(x) \Eta(x ) + 16 x \Eta(x) \Eta(x ) - \Eta(x ) =0 # 8 for \Eta(x) we obtain # # [ +--------+ ] # | | 2 | # | -B +\|B -4 A C | # | - |1/8 # \Eta(i x)=| --------------- | \where # [ 2 A ] # # 4 8 # A = \Eta(x ) , # # 4 16 # B = 16 i x \Eta(x ) , # # # [ 4 3 ] # 2 24 | \Eta(x ) |24 # C = -\Eta(-x ) = -| ----------------- | # | 2 8 | # [ \Eta(x ) \Eta(x ) ] #