eta

Function: eta
Section: transcendental
C-Name: eta0
Prototype: GD0,L,p
Help: eta(z,{flag=0}): if flag=0, returns prod(n=1,oo, 1-q^n), where
 q = exp(2 i Pi z) if z is a complex scalar (belonging to the upper half plane);
 q = z if z is a p-adic number or can be converted to a power series.
 If flag is nonzero, the function only applies to complex scalars and returns
 the true eta function, with the factor q^(1/24) included.

Doc: Variants of \idx{Dedekind}'s $\eta$ function.
 If $\fl = 0$, return $\prod_{n=1}^\infty(1-q^n)$, where $q$ depends on $x$
 in the following way:

 \item $q = e^{2i\pi x}$ if $x$ is a \emph{complex number} (which must then
 have positive imaginary part); notice that the factor $q^{1/24}$ is
 missing!

 \item $q = x$ if $x$ is a \typ{PADIC}, or can be converted to a
 \emph{power series} (which must then have positive valuation).

 If $\fl$ is nonzero, $x$ is converted to a complex number and we return the
 true $\eta$ function, $q^{1/24}\prod_{n=1}^\infty(1-q^n)$,
 where $q = e^{2i\pi x}$.
Variant:
 Also available is \fun{GEN}{trueeta}{GEN x, long prec} ($\fl=1$).