Comparing Eqs.\ref{eq:Pade_Exp_Decay_22} and \ref{eq:expandDecay31} to Eq. \ref{eq:Pade_zero_index}  , we notice that in both expansions,  \(\kappa_{i,0}\) and \(\kappa_{i,2}\) equal zero and, in addition, \(\gamma_{i,q}>0\) for all \(q\).  This suggests that the Padé approximants for \(y_i\left(B\right)\) should likewise set the coefficients for the numerator in Eq. \ref{eq:Pade_zero_index}  equal to zero for all even powers of \(B\) and require the coefficients for the denominator to be positive values.  It further suggests that in addition to requiring  that  \(\kappa_{i,0}=\ 0\)\(\kappa_{i,2}=0\) , and \(\gamma_{i,0}=1\), we should set the initial values of the remaining coefficients in the numerator as follows: \(\kappa_{i,1}=\ -\left|a_i\ \right|\) ,  \(\kappa_{i,3}=\ -\frac{1}{60}\left|a_i\right|\) — where \(a_i\) represents the approximate asymptotic value of \(y_i\) at high field — and the initial values of the coefficients in the denominator as \(\gamma_{i,1}=\frac{1}{2}\),  \(\gamma_{i,2}=\frac{1}{10}\),  etc.
Before proceeding, however, we still need to account for the approximately linear term in \(B\) in the high field limit. One way to do this while still using the results of Eq. \ref{eq:expandDecay31} is  to drop the highest order term in the numerator. Our proposed function then becomes