Asymptotic behavior of Fresnel-like integral of an exponential

Question

Given the integral

$$ I(t) = \int_0^t \mathrm{d}x \exp(-[\alpha \cos x + \beta \sin x]),\quad \alpha,\beta\in \mathbb{R}, $$

how can one obtain the asymptotic behavior for $t \to \infty$?

Answer 1

Note that the integrand is $2\pi$-periodic. Hence, since

$$\int_0^{2\pi}e^{-\alpha\sin(x)+\beta\cos(x)}\,dx=2\pi \text{I}_0\left(\sqrt{\alpha^2+\beta^2}\right)$$

then we have for $t\in (2n\pi,2(n+1)\pi]$

$$\int_0^{t}e^{-\alpha\sin(x)+\beta\cos(x)}\,dx=2n\pi \text{I}_0\left(\sqrt{\alpha^2+\beta^2}\right)+\int_{2n\pi}^t e^{-\alpha\sin(x)+\beta\cos(x)}\,dx$$

Hence, we have asymptotically for large $t$

$$\int_0^t e^{-\alpha\sin(x)+\beta\cos(x)}\,dx\sim\text{I}_0\left(\sqrt{\alpha^2+\beta^2}\right) t$$