I have a function expressed as the ratio of two exponential series with certain parameters $$\frac{\sum\limits_{j=1}^{i-1} \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^{i-1} (b^j-b^l)}}{\sum\limits_{j=1}^{i} \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^{i} (b^j-b^l)}}$$ with $a, b $ positive and $r_j = \frac{1}{500} b^j$ which appears in solving a DE. Simplifying this ratio is difficult; however, we can fit this ratio in Mathematica to a nonlinear function of the form $c_1+\frac{c}{t}$, with $c_1$, $c$ as constants, which gives a good fit. The issue is that Mathematica gives numerical values for $c_1$, $c$, and I want these values to be expressed in terms of the parameters in the ratio above. The reason for this is that changing the values of the parameters will produce new numerical values of $c_1$, $c$. Is there a possible way to get $c_1$, $c$ in terms of $a$, $b$, $r_j$?
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$\begingroup$ It’s unlikely that there’s a closed form for the best fit. But if you replace $t$ with $1/u$ and approximate the ratio with a linear Taylor series around some value of $u$, you might be able to guess some $u$ which would give a good fit. $\endgroup$user44143– user441432022-01-06 13:43:15 +00:00Commented Jan 6, 2022 at 13:43
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$\begingroup$ I would think that, by its very nature, a fitting procedure is a numerical procedure; it makes little sense to try and capture that symbolically; things might be different if you are interested in the asymptotics in some limit (small $a$, small $b$). $\endgroup$Carlo Beenakker– Carlo Beenakker2022-01-06 14:59:00 +00:00Commented Jan 6, 2022 at 14:59
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2$\begingroup$ $$ \frac{\sum\limits_{j=1}^{i-1} \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^{i-1} (b^j-b^l)}}{\sum\limits_{j=1}^i \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^i (b^j-b^l)}} $$ I wonder whether people would consider the above easier to read if it were formatted like this: $$ \left. \sum_{j=1}^{i-1} \frac{\exp(-ar_jt)}{\prod_{\ell=1\\\ell \ne j}^{i-1} (b^j-b^\ell)} \right/ \sum_{j=1}^i \frac{\exp(-ar_jt)}{\prod_{\ell=1\\\ell \ne j}^i (b^j-b^\ell)} $$ $\endgroup$Michael Hardy– Michael Hardy2022-01-06 17:54:03 +00:00Commented Jan 6, 2022 at 17:54
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