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Is it possible to solve the following equation analytically for r:

$$A = exp\bigg[a\bigg(\frac{r}{r_{0a}}\bigg)^b\bigg] + exp\bigg[c\bigg(\frac{r}{r_{0c}}\bigg)^d\bigg] + exp\bigg[f\bigg(\frac{r}{r_{0f}}\bigg)^g\bigg] $$

Where every quantity is known except for r? I can't figure it out, and I don't think it is. Any idea how accurate a first order Taylor expansion would be?

ArcaneAnomaly
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  • 11 variables? Please. – user321120 Oct 17 '18 at 20:44
  • I think you are asking that, given real numbers $A, a, c,f, b, d, g$, find $r \in \mathbb{R}$ to solve $A=exp(ar^b) + exp(cr^d) + exp(fr^g)$. (Here I am consolidating some of your constants to make the problem simpler, without loss of generality.) I don't think it can be solved analytically. Taylor expansion may still be hard and likely inaccurate. But numerical methods should give accurate results, for example if all the exponentials are nondecreasing you can do a bisection search that has exponential convergence speed. You can get, say, 10 decimal places of accuracy easily. – Michael Oct 17 '18 at 20:52
  • Of course the signs of your constants are important. For example no solution exists if $A\leq 0$. Also, if $a,c,f \geq 0$ and $b,d,g$ are even integers then no solution exists if $A<3$. (If $b,d,g$ are not integers then I assume $r \geq 0$ is the assumed domain?) – Michael Oct 17 '18 at 20:59
  • Thank you for your help. A > 0, a, c, and f < 0, and none of the parameters are integers, so a solution definitely exists. – ArcaneAnomaly Oct 17 '18 at 21:11

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