The Riemann-Lebesgue Lemma Examples 1

# The Riemann-Lebesgue Lemma Examples 1

Recall from The Riemann-Lebesgue Lemma page that if $f \in L(I)$ then for all $\beta \in \mathbb{R}$ we have that:

(1)
\begin{align} \quad \lim_{\alpha \to \infty} \int_I f(t) \sin (\alpha t + \beta) \: dt = 0 \end{align}

We will now look at some example problems regarding the Riemann-Lebesgue lemma due to its significance (as we will see later).

## Example 1

Use the Riemann-Lebesgue lemma to evaluate $\displaystyle{\lim_{\alpha \to \infty} \int_0^2 (t^2 + \cos t)(1 + \cos (\alpha t + 2003)) \: dt}$.

We have that:

(2)
\begin{align} \quad \lim_{\alpha \to \infty} \int_0^2 (t^2 + \cos t)(1 + \cos (\alpha t + 2003)) \: dt &= \lim_{\alpha \to \infty} \left [ \int_0^2 t^2 \: dt + \int_0^2 t^2 \cos (\alpha t + 2003) \: t + \int_0^2 \cos t \: dt + \int_0^2 \cos t \cos (\alpha t + 2003) \: dt \right ] \end{align}

Notice that $t^2, \cos t \in L([0, 2])$, so by the Riemann-Lebesgue lemma we have that $\displaystyle{\lim_{\alpha \to \infty} \int_0^2 t^2 \cos (\alpha t + 2003) \: dt =0}$ and $\displaystyle{\lim_{\alpha \to \infty} \int_0^2 \cos t \cos (\alpha t + 2003) \: dt = 0}$, thus:

(3)
\begin{align} \quad \lim_{\alpha \to \infty} \int_0^2 (t^2 + \cos t)(1 + \cos (\alpha t + 2003)) \: dt &= \int_0^2 t^2 \: dt + \int_0^2 \cos t \: dt \\ &= \frac{t^3}{3} \biggr \lvert_0^2 + \sin t \biggr \lvert_{0}^{2} \\ &= \frac{8}{3} + \sin (2) \end{align}