$\int_{-\pi}^{\pi} e^{x \sin(t-\tau)} d\tau$

Question

I am working on a problem that involves the integral

$$\int_{-\pi}^{\pi} e^{x \sin(t-\tau)} d\tau$$

where $$t$$ is a parameter and $$x$$ is a constant. This integral appears in a specific context of my research related to Electromagnetic. I am looking for ways to derive this integral, either to find a closed-form solution or a simplification that could be more easily evaluated numerically.

Background

The integral is intriguing because it combines exponential functions with the trigonometric sine function, where the sine function includes a difference in its argument. The bounds of the integral cover a full period of the sine function, which suggests that there might be symmetrical properties or periodicity that could be exploited in the derivation.

Attempted Approaches

1. Direct Integration: I initially attempted to integrate directly using standard techniques but did not arrive at a simplification that was helpful.

2. Change of Variable: Considering a change of variable such as $$\theta = t - \tau$$ did not significantly simplify the integration, possibly due to my approach in handling the limits of integration or the periodic nature of the sine function.

3. Fourier Series and Euler's Formula: I also considered decomposing the sine function using Euler's formula or exploring Fourier series representations, thinking it might reveal a more manageable form for integration. However, I'm unsure of how to proceed effectively in this direction.

Questions

1. Has anyone encountered a similar integral, and how did you approach its derivation?

2. Are there specific change of variable techniques or trigonometric identities that would simplify this integral?

3. Could Fourier series or Euler's formula provide a pathway to either a closed-form solution or a more accessible numerical evaluation?

4. Any recommendations on literature or resources that tackle similar integrals would be highly appreciated.

Thank you for any insights or guidance you can provide!

• $$\int_{-\pi}^{\pi} e^{x \sin(t-\tau)} d\tau = \int_{-\pi}^{\pi} e^{x \sin(\tau)} d\tau=2\pi I_0(x)$$ Feb 10 at 12:57
• @Quanto thanks, should not be a function of t? Feb 10 at 13:03
• I think it is not bessel exactly, becauae after change of veriable the integral limits doesnot match with bessel. Feb 10 at 13:38
• @AlirezaGhazavi. The answer is correct: $2 \pi I_0(x)$ .Solution is independent of the parameter t. Feb 10 at 16:02
• @AlirezaGhazavi: notice that $$\int^\pi_{-\pi}e^{x\sin t}\,dt= \int^{\pi/2}_{-3\pi/2}e^{x\sin(t+\pi/2)}\,dt=\int^{\pi/2}_{-3\pi/2}e^{x\cos t}\,dt=\int^{\pi}_{-\pi}e^{x\cos t}\,dt$$ The last equality follows from periodicity. Feb 10 at 16:28