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?

  • $\begingroup$ 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
  • 1
    $\begingroup$ 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


Try the following

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.