Question:

I want to evaluate the decay estimate of the integral

$I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $

for sufficiently large $ t $, where $ d \in \mathbb{Z}_{\ge 2} $ and $ v \in \mathbb{R}_{\ge 0}$.

It is hypothesized that the inequality

$ |I^d(t; v)| \le C t^{-\frac{d-1}{2}} $

may hold with constant $C$.

How can I evaluate this estimate?

Question:

I want to evaluate the decay estimate of the integral

$I^d(t; v) = \int_0^{\sqrt{d}\pi} dr \, r^{d-2} \int_0^\pi \sin(tr) e^{i\sqrt{d}vtr\cos\theta} \sin^{d-2}\theta \, d\theta $

for sufficiently large $ t $, where $ d \in \mathbb{Z}_{\ge 2} $ and $ v \in \mathbb{R}_{\ge 0}$.

It is hypothesized that the inequality

$ |I^d(t; v)| \le C t^{-\frac{d-1}{2}} $

may hold with constant $C$.

How can I evaluate this estimate?

update

The integral is a solution for discrete linear wave equation. I am interested in parallel structures between continuous and discrete system.