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?
1 Answer
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Try the following
Series[f[InverseFunction[g][y]],{y,0,10}]
f[x_] = Sin[x]; g[x_] = 3 + x^(1/2); sol = First@Solve[g[x] == y, x]
yiels{x -> 9 - 6 y + y^2}
andSeries[f[x /. sol], {y, 0, 2}]
yieldsSeriesData[y, 0, { Sin[9], (-6) Cos[9], Cos[9] - 18 Sin[9]}, 0, 3, 1]
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