EtaQuotient(C, L)

qetadom.spad line 113

EtaQuotient implements the (multiplicative) group of eta-functions in their expansion at infinity. Elements can be represented as Laurent series in q with a prefactor of q^(n/24), (n=0,…,23). Note that this domain keeps the “fractional part” always separate from the “series part” even when the fractional part is an integer.

1: %
from MagmaWithUnit
*: (%, %) -> %
from Magma
/: (%, %) -> %
from Group
=: (%, %) -> Boolean
from BasicType
^: (%, Integer ) -> %
from Group
^: (%, NonNegativeInteger ) -> %
from MagmaWithUnit
^: (%, PositiveInteger ) -> %
from Magma
~=: (%, %) -> Boolean
from BasicType
coerce: % -> OutputForm
from CoercibleTo OutputForm
coerce: C -> %
coerce(c) returns an element of this domain corresponding to c.
commutator: (%, %) -> %
from Group
conjugate: (%, %) -> %
from Group
eta: PositiveInteger -> %
eta(n) returns q^(n/24)*prod_{k=1}^infty (1-q^{kn}).
etaPower: (PositiveInteger , Integer ) -> %
etaPower(d, e) returns eta(d)^e.
etaQuotient: (List Integer , List Integer ) -> %
etaQuotient(divs, r) returns the $eta$-quotient product(eta(divs.i)^(r.i),i=1..#divs). It is assumed that the lengths of the input lists are equal.
hash: % -> SingleInteger
from SetCategory
hashUpdate!: (HashState , %) -> HashState
from SetCategory
inv: % -> %
from Group
latex: % -> String
from SetCategory
leftPower: (%, NonNegativeInteger ) -> %
from MagmaWithUnit
leftPower: (%, PositiveInteger ) -> %
from Magma
leftRecip: % -> Union(%, failed)
from MagmaWithUnit
one?: % -> Boolean
from MagmaWithUnit
prefactor: % -> NonNegativeInteger
prefactor(x) returns the exponent times 24 of the fractional q power of x where q is the variable of L. The returned value is in the range 0..23, because integer powers of q are moved to the series part.
q24: Integer -> %
q24(n) returns q^(n/24).
recip: % -> Union(%, failed)
from MagmaWithUnit
retract: % -> L
retract(x) aborts with error if prefactor is not divisible by 24. Otherwise it returns q^r*series(x) where r=prefactor(x)/24 and q is the variable of L.
rightPower: (%, NonNegativeInteger ) -> %
from MagmaWithUnit
rightPower: (%, PositiveInteger ) -> %
from Magma
rightRecip: % -> Union(%, failed)
from MagmaWithUnit
sample: %
from MagmaWithUnit
series: % -> L
series(x) returns the Laurent series part of x. x = q24(prefactor x)*series(x)
toEta: Polynomial C -> %
toEta(p) assumes that all variables are of the form Ei with the letter E and a positive number i. In the polynomial p the powers Ei^ni will be replaced by eta(i)^ni. If l is the leading monomial p, then it is assumed that prefactor(toEta(l))=prefactor(toEta(m)) for every monomial m of p. If

BasicType

CoercibleTo OutputForm

Group

Magma

MagmaWithUnit

Monoid

SemiGroup

SetCategory

TwoSidedRecip

unitsKnown