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I was playing around with trig functions and came across the Dottie Number $D \approx 0.73$ as the root of $f(x) = \cos{x}-x$. Then some time later I came across a similar function: $g(x) = \sqrt2\:\cos{x} - x$, its root is about $0.89$. I wondered if there was any way to write this new number $H \approx 0.89$ in terms of $D$ (Dottie's Number) and if there was a formula that allows us to, if given the root of any function $j_0(x) = k(x) + l(x)$, use it to find the root of a function $j_1(x) = a\cdot k(x) + l(x)$ for any $a\in \mathbb{C}$

If this doesn't work for all functions it would be good to know for what kind of functions a formula for this exists.

Would really appreciate the help!

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  • $\begingroup$ There is a series solution for the solution of $\cos(x)=ax$ here, so an equivalent problem is writing the series in terms of one another. This depends on writing $J_n(-\frac nb)$ in terms of $J_n(-\frac n{\sqrt 2})$ with the Bessel J function $\endgroup$
    – Тyma Gaidash
    Commented Nov 24, 2023 at 23:47
  • $\begingroup$ @ТymaGaidash Interesting, I haven't worked with Bessel Functions before but I'll look into it, thanks for the help mate! $\endgroup$
    – hefe
    Commented Nov 25, 2023 at 13:43

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