EtaQuotientGamma(C, mx, CX, xi, LX)ΒΆ
- C: Join(Algebra Fraction Integer , IntegralDomain )
- mx: PositiveInteger
- CX: Algebra C
- xi: CX
- LX: UnivariateLaurentSeriesCategory CX
EtaQuotientGamma(C
, mx
, CX
, xi
, LX
) represents the semigroup of eta-quotient expansions at gamma. The eta quotient need not be a modular function, but the (c*tau+d) factor is always ignored.
- =: (%, %) -> Boolean
- from BasicType
- ~=: (%, %) -> Boolean
- from BasicType
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- etaQuotient: SymbolicEtaQuotientGamma -> %
etaQuotient(s)
represents the expansion ofg_r
(gamma tau) in terms ofx
= exp(2 pii
tau/w) where w=width(m
,c
),r
= exponents(s
), andg_r
is defined by $g_r
(tau) = prod_{delta in divisors(s
)} eta(delta gamma tau)^{r_
{delta}}$.
- eulerExpansion: % -> LX
eulerExpansion(e)
returns the series expansion ofe
without any prefactor. Thetrue
series expansion (in terms of fractionalx
powers ofe
is given bylc
*x^p
*s
wherelc
= leadingCoefficiente
,x
= exp(2 pii
tau/w) where w=width(m
,c
),s
= symbolicEtaQuotiente
,c
= gamma(s
)(2,1),p
= (xExponents
) / 24,s
= eulerExpansione
.
- eulerFunctionPower: (PositiveInteger , NonNegativeInteger , Integer ) -> LX
eulerFunctionPower(u, v, p)
computes eulerFunction(1)^p
and replaces q=monomial(1,1)$LC
by monomial(xi^v,u
)$LX
- expansion: % -> LX
expansion(e)
should only be called if modular?(e
) holds, otherwise it might return an error. expansion(e
) returns the series expansion ofe
in terms ofx
= exp(2 pii
tau/w) where w=width(m
,c
),c
= gamma(symbolicEtaQuotiente
)(2,1).- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState , %) -> HashState
- from SetCategory
- latex: % -> String
- from SetCategory
- leadingCoefficient: % -> CX
leadingCoefficient(x)
returns the coefficient corresponding to the smallestq
-power with non-zero coefficient.
- modular?: % -> Boolean
modular?(x)
returnstrue
if the eta quotient is a modular function forGamma0
(nn
) where nn=level(symbolicEtaQuotientx
).
- substituteVariable: (PositiveInteger , NonNegativeInteger ) -> Record(k: Integer , c: C) -> Record(k: Integer , c: CX)
- Is the function [
k
,c
] +-> [u*k, xi^v*c]
- symbolicEtaQuotient: % -> SymbolicEtaQuotientGamma
symbolicEtaQuotient(x)
returns meta-data corresponding tox
.