I'm interested in whether there are any closed-form representations of $$ \int_0^{2\pi} e^{(\cos{t})^2 + k \cos{t}} dt \quad \text{ or } \quad \int_0^{2\pi} e^{(q+\cos{t})^2} dt $$ in terms of other special functions, where $k$ and $q$ are real constants. Without the quadratic term I know the first integral is a (scaled) modified Bessel function, but it seems like the quadratic case is much less common, and indeed, Mathematica yields no leads on possible functions involved. Suggestions on possible avenues of approach are appreciated!

  • $\begingroup$ Modified Bessel functions of the first kind are pretty much defined in terms of those integrals. $\endgroup$ – Jack D'Aurizio Apr 7 '18 at 16:04
  • $\begingroup$ @JackD'Aurizio Please explain. I have not seen any integral form for a Bessel function that deals with quadratic functions of cosine. $\endgroup$ – sourisse Apr 7 '18 at 16:18
  • $\begingroup$ $$\cos^2(x) = \frac{\cos(2x)+1}{2}$$ $\endgroup$ – Jack D'Aurizio Apr 7 '18 at 16:20
  • $\begingroup$ @JackD'Aurizio I realize this, but don't see how it permits evaluation of the integral above when $k$ is nonzero. $\endgroup$ – sourisse Apr 7 '18 at 16:23
  • $\begingroup$ You know the Fourier series of $e^{\cos^2}$, you know the Fourier series of $e^{k\cos}$, hence you know those integrals in terms of values of Bessel functions. $\endgroup$ – Jack D'Aurizio Apr 7 '18 at 16:26

Expanding the comments above: $$ e^{\cos^2\theta} = \sqrt{e}\,I_0\left(\tfrac{1}{2}\right)+2\sqrt{e}\sum_{n\geq 1}I_n\left(\tfrac{1}{2}\right)\cos(2n\theta)\tag{1} $$ $$ e^{k\cos\theta} = I_0(k)+2\sum_{n\geq 1}I_n(k)\cos(n\theta)\tag{2} $$ hence by the orthogonality relations in $L^2(0,2\pi)$ $$ \int_{0}^{2\pi}\exp\left(\cos^2\theta+k\cos\theta\right)\,d\theta = 2\pi\sqrt{e}\,I_0\left(\tfrac{1}{2}\right)I_0(k)+4\pi\sqrt{e}\sum_{n\geq 1}I_n\left(\tfrac{1}{2}\right)I_{2n}(k).\tag{3}$$

  • $\begingroup$ Neat, thank you! $\endgroup$ – sourisse Apr 8 '18 at 18:07

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