Finding Euler spiral convergence point for powers other than 2

Question

I'm playing around with Euler spirals, using a variable power to see how the plot changes,

$ \left\{\begin{matrix} x = \int_0^s \cos \left ( s^n \right ) ds \\ y = \int_0^s \sin \left ( s^n \right ) ds \end{matrix}\right. $

and finding the values for powers 2 or 3 work as expected in Mathematica:

Limit[ Integrate[Cos[s^2], {s, 0, x}], x -> ∞]
Limit[ Integrate[Sin[s^2], {s, 0, x}], x -> ∞]

$ \frac{\sqrt{\frac{\pi}{2}}}{2} \\ \frac{\sqrt{\frac{\pi}{2}}}{2} $

and similarly,

Limit[ Integrate[ Cos[s^3], {s, 0, x}], x -> ∞]
Limit[ Integrate[ Sin[s^3], {s, 0, x}], x -> ∞]

$ \frac{Gamma\left[\frac{1}{3} \right]}{2\sqrt{3}} \\ \frac{1}{6} Gamma \left [ \frac{1}{3} \right ] $

Plotting the spiral at different values (using a different general purpose graphics language) for $n$ suggests that there there should certainly be convergence values for both $x$ and $y$ for any value $1 \lt n \leq 3$ (and presumably above), but trying some of these leads to Mathematica giving most unhelpful answers:

$ Limit \left [ \int_0^s Cos \left [ s^{1.5} \right ]ds,s\rightarrow\infty \right ] \Rightarrow 0.451373 \\ Limit \left [ \int_0^s Sin \left [ s^{1.5} \right ]ds,s\rightarrow\infty \right ] \Rightarrow ComplexInfinity $

Which looks like the right $x$ coordinate, but a plain wrong $y$ coordinate, which quite clearly exists somewhere around 0.76ish:

Trying $n=2.5$ fails even harder, yielding "ComplexInfinity" for both coordinates, which is clearly incorrect (being somewhere near $(0.72,0.533)$):

Is there a way to make Mathematica generate these values anyway?

Or, is there a way to compute the general solution for symbolic power $n$ and getting a general formula for when $n\gt1$, using something like this (but then actually yielding a result):

$ Limit \left [ \int_0^s Cos \left [ s^n \right ]ds,s\rightarrow\infty \right ] \\ Limit \left [ \int_0^s Sin \left [ s^n \right ]ds,s\rightarrow\infty \right ] $

Answer 1

The problem occurs because of apparently complex form of the expression beyond the integral of Sin[s^k]. Nevertheless it is not too harmful to proceed. Adequate limits are real and we don't need playing to simplify complex expressions to explicitly real forms.

We can calculate the both integrals with appropriate assumptions:

f[x_, k_] = Integrate[{ Cos[s^k], Sin[s^k]}, {s, 0, x}, 
                      Assumptions -> {k > 1, x > 0}];

see it in TraditionalForm:

f[x, k] // TraditionalForm

and find appropriate limits with a needed assumption:

lim[k_] = Limit[ f[x, k], x -> Infinity, Assumptions -> k > 1]

{Cos[Pi/(2 k)] Gamma[1 + 1/k], Gamma[1 + 1/k] Sin[Pi/(2 k)]}

Now we can see that indeed limits are finite for k > 1. Here we write down a few exact values as expected:

Table[{k, lim[k]} // ComplexExpand // FullSimplify // ToRadicals, 
  {k, {5/4, 3/2, 2, 5/2, 3, 10/3}}] // TraditionalForm

We can plot adequate curves, e.g. using Quiet to avoid unneeded warnning related to apparently complex functions (involving I ), nevertheless if one uses f[x,k] it is not necessary.

With[{k = 5/2},
  ParametricPlot[f[x, k], {x, 0, 5}, PlotRange -> {{0.55, 1}, {0.3, 0.75}}, 
    PlotStyle -> {Black, Thickness[0.0015]}, AspectRatio -> Automatic, 
    PlotTheme -> "Detailed", Epilog -> {Red, PointSize[0.015], Point[lim[k]]}]]

and find slightly improved numerical limit:

lim[2.5]//Chop

{0.717812, 0.521521}