QEtaOneOverPi CΒΆ
oneoverpi.spad line 107 [edit on github]
QEtaOneOverPi provides functions to find formulas for the computation of 1/pi.
- cqD: (C, QEtaLaurentSeries C) -> QEtaLaurentSeries C
cqD(c,x)
returns monomial(c
,1)*D
(X
).
- expTruncated: (PositiveInteger, C) -> C if C has Field
expTruncated(s,x)
computes the exponential series truncated at powers
at the valuex
, i.e. it computes sum(x^n/factorial(n
),n=0
..s
).
- moebiusTransform: (Matrix Fraction Integer, C) -> C if C has Field
moebiusTransform(m, c)
computes the Moebius transform ofc
.
- moebiusTransform: (Matrix Integer, C) -> C if C has Field
moebiusTransform(m, c)
computes the Moebius transform ofc
.
- theta2ConstantQIncompleteH: (NonNegativeInteger, C) -> C if C has Field
theta2ConstantQIncompleteL(
m
,q
) computes $t_
{2h}(m
,q
)$ according to eqref{eq:t2
-m
-q
}, i.e. $t_
{2h}(m
,q
)}=t_
{2l}(m
,q
)+frac{2q^{frac{(m+1
)(m+2
)}{2}}}{1-q
}$.
- theta2ConstantQIncompleteL: (NonNegativeInteger, C) -> C if C has Field
theta2ConstantQIncompleteL(m, q)
computes $t_
{2l}(m
,q
)$ according to eqref{eq:t2
-m
-q
}, i.e. $t_
{2l}(m
,q
)}=2
sum_{n=0
}^mq^{ffrac{n
(n+1
)}{2}}$.
- theta3ConstantH: (NonNegativeInteger, C) -> C if C has Field
theta2ConstantQ4H(
m
,q
) computes an upper estimation of $thata_3(0,tau)$ ($q=
exp(pii
tau)$) for positive $q
$. It returns theta3ConstantL(m
,q
)+2*q^e/(1-q
) were $e=(m+1
)^2
$. It corresponds to $t_
{3h}(m
,q^2
)$ according to eqref{eq:t3
-m
-q
} where $t_
{3h}(m
,q
)=t_
{3l}(m
,q
)+frac{2q^{frac{(m+1
)^2
}{2}}}{1-q
}$
- theta3ConstantL: (NonNegativeInteger, C) -> C if C has Field
theta3Constant(
m
,q
) computes the truncation of the $theta_3(0,tau)$ series after them
-th term, i.e. $1 + 2 sum_{n=1
}^m
q^
{n^2
}$ ($q=
exp(pii
tau)$). It corresponds to $t_
{3l}(m
,q^2
)$ as defined in eqref{eq:t3
-m
-q
} where $t_
{3l}(m
,q
)=1+2
sum_{n=1
}^mq^{frac{n^2
}{2}}$.
- toQHTTL: (XHashTable(Matrix Integer, List Integer), XHashTable(Matrix Integer, Integer)) -> XHashTable(Matrix Integer, QEtaTruncatedLaurentSeries C)
toQHTTL(hl,estord)
converts each series of the input format into a truncated Laurent series. The indices (cusp matrices) are the same as in the input.
- transformTau: (QEtaLaurentSeries C, Matrix Integer, Fraction Integer) -> QEtaLaurentSeries C
transformTau(s,trf,w)
scales the series $s
$ (as a series in $q=
exp(pii
tau)$ by transformation $taumapstofrac{atau}{d
}$ (where trf is given by $\mt
{a}{0}{0}{d
}$) in the variable $q^
{frac{1}{w
}}$.
- uIPiCoefficient: (PositiveInteger, Matrix Integer, C, Matrix Integer, C) -> C if C has Field
uIPiCoefficient(nn, gammaN, tauN, gammay, tauy)
returns left( frac{N
c_N
(c_N
tau_N+d_N) det(gamma_y)}{(c_ytau_y+d_y)^2
} -frac{u_N c_y}{c_ytau_y+d_y} right). It corresponds to formula ref{eq:def-u
-pi
} without the w/i multiplier.
- uNCoefficient: (PositiveInteger, Matrix Integer, C) -> C
uNCoefficient(nn, gamma, tau)
returns determinant(gamma
) -nn*
(c*ta+d)^2
where gamma=matrix[[a,b
],[c
,d
]]. It corresponds to formula ref{eq:def-u
-N
}.