Inverse function of a taylor expansion

Question

If I have this equation

$$r(t)=\frac{1}{2}+\frac{t}{12}-\frac{t^3}{720}+\mathcal{O}(t^5)$$

Is there any possibility to find $t(r)$?

Answer 1

Using a suitable Computer Algebra System such as Mathematica to invert the powers series gives $t(r) = 12(r-\frac12) +\frac{144}5(r-\frac12)^3+\mathcal{O}(r-\frac12)^5\;$-- the best answer given the original information. The Mathematica command is $\texttt{InverseSeries[ 1/2 + t/12 - t^3/720 + O[t]^5, r]}$.

But this example is so simple you don't need a CAS to solve it. It is easy to see that the answer is $t(r)=c_1(r-\frac12) +c_3(r-\frac12)^3+\mathcal{O}(r-\frac12)^5\;$ for some constants $c_1,c_3.\;$ Now from the given equation we get $\;r(t)-\frac12 = \frac{t}{12} - \frac{t^3}{720}+\mathcal{O}{(t^5)},\;$ and so $\;12(r(t)-\frac12) = t - \frac{t^3}{60}+\mathcal{O}{(t^5)}\;$ and $c_1=12.\;$ Similarly, one more step gives $\;c_3=12^3\frac1{60}=\frac{144}5.$