Large Limits Approximations

When cast as above and with $u \gg 1$, a two term asymptotic approximation may be used with great accuracy³. We aptly call this the large limits approximation.

$$C(u) \approx \frac{1}{2} + \frac{1}{\pi u} \sin\left(\frac{\pi}{2} u^2\right)$$ (3)

$$S(u) \approx \frac{1}{2} - \frac{1}{\pi u} \cos\left(\frac{\pi}{2} u^2\right)$$ (4)

In figure 1 below, we show $C(u)$ and $S(u)$, along with the large limits approximations in equations 3 and 4. For visual clarity, we plot these approximations (and their negative counterparts) for $\vert u\vert>0$ even though they are intended to be valid only for $\vert u\vert \gg 1$.

In figure 2, we show the magnitude of the fractional errors resulting from use of the large limits approximation. We note that the two error functions are loosely anti-correlated, with larger errors corresponding to analogous portions of their $90^\circ$-out-of-phase sinusoids. Since the fractional error becomes infinite as $u$ approaches zero, we show a zoom-in of the plot.

Figure 1: C(u) and S(u) (solid) with large limits approximations (dashed)

Figure 2: Fractional Errors in C(u) and S(u) resulting from the large limits approximations.