What is the inverse function of $ y=x^3+x$? thanks
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2Do you really need its inverse? – Git Gud Dec 08 '13 at 12:47
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According to Wolfram Mathematica, the equation $x^3+x=y$ is solved by $$ x=\frac{\sqrt[3]{\sqrt{3} \sqrt{27 y^2+4}+9 y}}{\sqrt[3]{2} 3^{2/3}}-\frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{\sqrt{3} \sqrt{27 y^2+4}+9 y}}. $$
Siminore
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Proof by wolfram alpha. Did you at least verify that it is actually correct? – Tim Seguine Dec 08 '13 at 12:53
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@nim it isn't my answer. There is nothing wrong with it as far as I can see, I was only criticizing the presentation. – Tim Seguine Dec 08 '13 at 13:17
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Of course Mathematica can handle standard computations much better than I would :-) – Siminore Dec 08 '13 at 13:28
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I get $x= y-y^3+3y^5-12y^7+55y^9-273y^{11}+-...$ where the coefficients are given by http://oeis.org/A001764 with alternating signs. This is of course convergent up to the point where the closed form solution has its poles. The reference shows that, as a generating function, the coefficients have nice closed forms in terms of binomial coefficients.
R. J. Mathar
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2This isn't even convergent, is it? I am not convinced a series solution is so useful when closed form analytic solutions are possible. – Tim Seguine Dec 08 '13 at 13:47