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I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me that complex analysis might lend some clarity, maybe the rewrite: $$\int(\cos{x})^xdx=\int\dfrac{(e^{i*x}+e^{-i*x})^x}{2^x}dx$$ would point me in the correct direction, but analytically I'm still stumped here.

Does anyone have any knowledge of this family of equations? The integral probably can't be analytically evaluated with the maths we have today, but any information about the function type would be appreciated.

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  • $\begingroup$ I am pretty sure this integral does not have any solution in terms of standard mathematical functions. $\endgroup$ – Loreno Heer Aug 28 '15 at 0:13
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    $\begingroup$ Even defining your integrand is problematic whenever $\cos x < 0$. And even if you restrict yourself to intervals where this doesn't happen, you'll get singularities at the endpoints when $x$ is negative... $\endgroup$ – Micah Aug 28 '15 at 0:16
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    $\begingroup$ Just quickly looking ta the graph: Note that the function is not defined continuously in $\left[\frac{(4k+1)\pi}{2}, \frac{(4k+3)\pi}{2}\right]$, which is where $\cos x \lt 0$. $\endgroup$ – Lukas Juhrich Aug 28 '15 at 0:16
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Yes, $\cos(x)^x$ is not hard to "explore". I don't know if it's been the subject of any published work, other than OEIS sequence A215347. It satisfies, for example, the differential equation $$ y y'' = (y')^2 - (2 \tan(x) + x \sec^2(x)) y^2$$ and it has the Maclaurin series

$$ y = 1 - \dfrac{x^3}{2} - \dfrac{x^5}{12} + \dfrac{x^6}{8} - \dfrac{x^7}{45} + \dfrac{x^8}{24} - \dfrac{139}{5040} x^9 + \ldots $$

As for being defined, $(\cos x)^x = \exp(x \ln \cos x)$. This will be multivalued on the complex plane with logarithmic branch points where $\cos x = 0$.

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  • $\begingroup$ Does that ODE appear somewhere in an applied field? $\endgroup$ – DLV Aug 28 '15 at 0:41
  • $\begingroup$ @David : Probably not. $\endgroup$ – Robert Israel Aug 28 '15 at 0:48
  • $\begingroup$ The fresnel integral is what led me to this problem, but that's used to evaluate: $$\int \cos (x^2) dx$$ $\endgroup$ – Alec Rhea Aug 28 '15 at 2:27
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    $\begingroup$ Which is applied in light-wave mechanics through changing refractive indexes - like sunlight from air to glass. $\endgroup$ – Alec Rhea Aug 28 '15 at 2:30

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