Infinite Limits Properties and Approximations

By observation of the above approximation for $C(u)$ and $S(u)$, we see that as $u$ goes to infinity, the second term in each approximation goes to zero. Since both $C(u)$ and $S(u)$ are odd symmetric, we then have the following convenient facts:

$$C(\infty) = S(\infty) = \frac{1}{2}$$ (5)

$$C(-\infty) = S(-\infty) = -\frac{1}{2}$$ (6)

When making use of equations 5 and 6 for large but finite limits, we will use the term ``infinite limits approximations.'' These equalities are in fact considered properties of Fresnel integrals, as is

$$C(0) = S(0) = 0.$$ (7)

These will prove useful in our proof below.