Show $\int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi$

Question

I am tring to prove $$ \int_{-\infty}^\infty\int_0^\infty \exp(\cos t-1-t^2) \cos(\sin t-t x)\,\mathrm dt\mathrm dx=\pi. $$ Numerical integration in Mathematica (truncating the integration bounds on $x$ to $(-30,30)$ shows agreement with the conectured exact value of $\pi$.

Interchanging the order of integration is not applicable here because $\int_{-\infty}^\infty\cos(\sin t-tx)\,\mathrm dx$ does not converge. I also do not see a useful $u$-substitution to simplify things. I can write the entire integrand as a single (complicated) exponential function using $\cos(\sin t-tx)=\Re \exp(i(\sin t-tx))$ but this too has not produced any fruit. Also the exponential term $\exp(\cos t-t-t^2)$ is critical to the convergence of the integral so expanding it as a Taylor series and integration termwise won't work. Perhaps someone sees a path forward using complex methods.

Answer 1

Let $I$ denote the integral. Then

\begin{align} I &= \frac{1}{2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(\cos t - 1 - t^2) \cos(\sin t - t x) \, \mathrm{d}t\mathrm{d}x \tag{1} \\ &= \frac{1}{2} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(e^{it}-t^2-ixt-1) \, \mathrm{d}t\mathrm{d}x \tag{2} \\ &= \frac{1}{2} \int_{-\infty}^{\infty} \sum_{n=0}^{\infty} \frac{1}{n!} \int_{-\infty}^{\infty} e^{int}\exp(-t^2-ixt-1) \, \mathrm{d}t\mathrm{d}x \tag{3} \\ &= \frac{1}{2} \int_{-\infty}^{\infty} \left( \sum_{n=0}^{\infty} \frac{1}{n!} \sqrt{\pi} \, e^{-\frac{1}{4} (n-x)^2-1} \right) \mathrm{d}x \tag{4} \\ &= \frac{\sqrt{\pi}}{2e} \sum_{n=0}^{\infty} \frac{1}{n!} \int_{-\infty}^{\infty} e^{-\frac{1}{4} (n-x)^2} \mathrm{d}x \tag{5} \\ &= \frac{\sqrt{\pi}}{2e} \sum_{n=0}^{\infty} \frac{1}{n!} (2\sqrt{\pi}) \tag{6} \\ &= \pi. \end{align}

Here are some explanations:

• $\text{(3)}$ : Fubini's theorem is applicable: $$ \int_{-\infty}^{\infty} \sum_{n=0}^{\infty} \left| \frac{e^{int}}{n!} \exp(-t^2-ixt-1) \right| \, \mathrm{d}t = \int_{-\infty}^{\infty} \sum_{n=0}^{\infty} \frac{1}{n!} \exp(-t^2-1) \, \mathrm{d}t < \infty. $$

• $\text{(4)}$, $\text{(6)}$ : We have $$\int_{-\infty}^{\infty} e^{-a(x-z)^2} \, \mathrm{d}x = \sqrt{\frac{\pi}{a}}$$ for any $a > 0$ and $z \in \mathbb{C}$. Now, use this and complete the square of the exponent to conclude.

• $\text{(5)}$ : Tonelli's theorem is applicable.