How do I find Maclaurin-like series for parametrically defined curves?

Question

Suppose I have a parametrically defined curve: $$x=x(t),\quad y=y(t).$$ For some parametrically defined curves, such as $x=t^2,y=\sin(t^2)$ (a very basic example, I know) we can eliminate $t$ and write the equation of the curve in terms of $x$ and $y$ only.

However, for many other parametrically defined curves, we cannot eliminate $t$. I have been wondering if there's a way we could write the equation of the curve as an infinite Maclaurin-like series instead, ie as $$y=\sum_{n=0}^{\infty}a_n x^n$$ where $a_n$ is the coefficient of $x^n$.

Do you know of any way we could do this?

I know you would like me to show any work I have done myself but I simply have no idea where to start.

Thank you for your time and help.

Answer 1

In principle, this should be doable for truncated series.

Expand $x$ as a Taylor series of $t$. Make a series reversion to get $t$ as a function of $x$. Plug the resulting expression in $y$ and expand $y$ as a Taylor series of $x$.

Let us try for $$x=\log \left(t+e^{-t}\right)\qquad \text{and} \qquad y=\sin(t^3)$$ $$x=\frac{t^2}{2}-\frac{t^3}{6}-\frac{t^4}{12}+\frac{3 t^5}{40}+\frac{t^6}{120}-\frac{31 t^7}{1008}+O\left(t^8\right)$$ $$t= \sqrt{2x}+\frac{x}{3}+\frac{11 x^{3/2}}{18 \sqrt{2}}+\frac{49 x^2}{270}+\frac{409 x^{5/2}}{2160 \sqrt{2}}+\frac{1189 x^3}{17010}+O\left(x^{7/2}\right)$$ $$y=2 \sqrt{2} x^{3/2}+2 x^2+\frac{13 x^{5/2}}{3 \sqrt{2}}+\frac{317 x^3}{135}+\frac{1147 x^{7/2}}{360 \sqrt{2}}+\frac{970 x^4}{567}+O\left(x^{9/2}\right)$$

Checking $$x=\log \left(t+e^{-t}\right)\implies t=W\left(-e^{-e^x}\right)+e^x$$ $$y=\sin \left(\left(W\left(-e^{-e^x}\right)+e^x\right)^3\right)$$ $$y=2 \sqrt{2} x^{3/2}+2 x^2+\frac{13 x^{5/2}}{3 \sqrt{2}}+\frac{317 x^3}{135}+\frac{1147 x^{7/2}}{360 \sqrt{2}}+\frac{1198 x^4}{405}+O\left(x^{9/2}\right)$$

Answer 2

Too long for a comment:

I might be mistaken, but it seems like if a global series representation for the curve exists, then it can't be a power series. It is known that every power series is the Taylor series of some function in $C^\infty$, so if $x$ and $y$ in the series equation you provided are not shorthand for $x(t)$ and $y(t)$, then it seems like a power series expansion can only represent the curve $x=x(t)$, $y=y(t)$ locally, specifically in "sections" where the curve can be thought of as the graph of a function. Thus, for instance, there is no sequence of real numbers $\{a_n\}_{n=0}^{\infty}$ such that for all points $(x,y)$ on the unit circle, we can write

$$y=\sum_{n=0}^{\infty}a_n x^n$$

However, if we restrict our attention to parts of the unit circle that are graphs of functions, namely its upper branch $y=\sqrt{1-x^2}$ and lower branch $y=-\sqrt{1-x^2}$, then we can obtain two series representations. Each can be found by using the Maclaurin series expansion of $\sqrt{1-x}$ evaluated at $x^2$.