Function: weber
Section: transcendental
C-Name: weber0
Prototype: GD0,L,p
Help: weber(x,{flag=0}): One of Weber's f function of x. flag is optional,
 and can be 0: default, function f(x)=exp(-i*Pi/24)*eta((x+1)/2)/eta(x),
 1: function f1(x)=eta(x/2)/eta(x)
 2: function f2(x)=sqrt(2)*eta(2*x)/eta(x). Note that
 j = (f^24-16)^3/f^24 = (f1^24+16)^3/f1^24 = (f2^24+16)^3/f2^24.
Doc: one of Weber's three $f$ functions.
 If $\fl=0$, returns
 $$f(x)=\exp(-i\pi/24)\cdot\eta((x+1)/2)\,/\,\eta(x) \quad\hbox{such that}\quad
 j=(f^{24}-16)^3/f^{24}\,,$$
 where $j$ is the elliptic $j$-invariant  (see the function \kbd{ellj}).
 If $\fl=1$, returns
 $$f_1(x)=\eta(x/2)\,/\,\eta(x)\quad\hbox{such that}\quad
 j=(f_1^{24}+16)^3/f_1^{24}\,.$$
 Finally, if $\fl=2$, returns
 $$f_2(x)=\sqrt{2}\eta(2x)\,/\,\eta(x)\quad\hbox{such that}\quad
 j=(f_2^{24}+16)^3/f_2^{24}.$$
 Note the identities $f^8=f_1^8+f_2^8$ and $ff_1f_2=\sqrt2$.
Variant: Also available are \fun{GEN}{weberf}{GEN x, long prec},
 \fun{GEN}{weberf1}{GEN x, long prec} and \fun{GEN}{weberf2}{GEN x, long prec}.