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We first relate an approximate $n^{th}$-order left derivative of $s(R, f^t)$at the F-pure threshold $c$ to the F-splitting ratio $r_F(R, f^c)$. Next, weapply the methods developed by Monsky and Teixeira in their investigation ofsyzygy gaps and $p$-fractals to obtain uniform convergence of the F-signaturewhen $f$ is a product of distinct linear polynomials in two variables. Finally,we explicitly compute the F-signature function for several examples usingMacaulay2 code outlined in the last section of this paper.

Relating F-Signature and F-Splitting Ratio of Pairs Using Left-Derivatives