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 power- sat the value- x, 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 of- c.
- moebiusTransform: (Matrix Integer, C) -> C if C has Field
- moebiusTransform(m, c)computes the Moebius transform of- c.
- 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. $- t_{2l}(- m,- q)}- =2sum_{- n=0}^mq^{ffrac{- n(- n+1)}{2}}$.
- theta3ConstantH: (NonNegativeInteger, C) -> C if C has Field
- theta2ConstantQ4H( - m,- q) computes an upper estimation of $thata_3(0,tau)$ ($- q=exp(pi- itau)$) 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 the- m-th term, i.e. $1 + 2 sum_{- n=1}- ^m- q^{- n^2}$ ($- q=exp(pi- itau)$). It corresponds to $- t_{3l}(- m,- q^2)$ as defined in eqref{- eq:t3-- m-- q} where $- t_{3l}(- m,- q)- =1+2sum_{- n=1}^mq^{frac{- n^2}{2}}$.
- 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 $- q=exp(pi- itau)$ 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_Ntau_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)- ^2where gamma=matrix[[a,- b],[- c,- d]]. It corresponds to formula ref{eq:def-- u-- N}.