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 s at 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 $thata_3(0,tau)$ ($q=exp(pi i tau)$) 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 the m-th term, i.e. $1 + 2 sum_{n=1}^m q^{n^2}$ ($q=exp(pi i tau)$). 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 $q=exp(pi i tau)$ 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_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)^2 where gamma=matrix[[a,b],[c,d]]. It corresponds to formula ref{eq:def-u-N}.