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Suppose I have a well-known function $f(x)$ and I want to know if this function can be expanded into power series of another function $g(x)$, like $f(x)=\sum_n a_n g^n(x)$, where $a_n$ is the coefficient of $n$-th power of $g(x)$. How should I do this using Mathematica?

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  • $\begingroup$ How about f[x_] = Sin[x]; g[x_] = 3 + x^(1/2); sol = First@Solve[g[x] == y, x] yiels {x -> 9 - 6 y + y^2} and Series[f[x /. sol], {y, 0, 2}] yields SeriesData[y, 0, { Sin[9], (-6) Cos[9], Cos[9] - 18 Sin[9]}, 0, 3, 1] . $\endgroup$
    – Akku14
    Commented Oct 16, 2020 at 19:12
  • $\begingroup$ Thanks a lot! It works! $\endgroup$
    – lol
    Commented Oct 16, 2020 at 22:12
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    $\begingroup$ For searching purposes: what is being sought here is sometimes referred to as a Bürmann series. $\endgroup$ Commented Jan 27, 2021 at 12:18

1 Answer 1

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Try the following

Series[f[InverseFunction[g][y]],{y,0,10}]
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  • $\begingroup$ This was very good to know, In general if this operation is timing out is there any way to tell InverseFunction, "hey, I only plan to use this in a taylor series, we don't need a complete inverse, just enough to get the next 10 terms" $\endgroup$ Commented Apr 25 at 3:17

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