Rectangular Window

We noted in the previous subsection that the large limits same sign approximation is not usable for $k=0$. This was, in fact, because when $k=0$ and $\alpha$ is small, $l_1$ and $l_2$ become too small to use the large limits approximation in any form. Fortunately, we may consider alternate functions that well approximate the Fresnel cosine and sine integrals for $\vert u\vert<1$. Specifically, the author determined empirically that equations 66 and 67 are well-modeled as

$$\int_{\pm l_1}^{\pm l_2} \cos\left(\frac{\pi}{2}u^2\right) du \approx l_2-l_1$$ (126)

$$% \int_{\pm l_1}^{\pm l_2} \sin\left(\frac{\pi}{2}u^2\right) du \approx \ensuremath{\frac{l_2^3}{2}}- \ensuremath{\frac{l_1^3}{2}}.$$ (127)

Continuing algebraically as before and considering $l_1$ and $1_2$ when $k=0$, we obtain:

$$\Re{Y_a(0)} \approx N$$ (128)

$$\Im{Y_a(0)} \approx \ensuremath{\frac{\alpha N^3}{4\pi}}.$$ (129)

The imaginary expression may be solved for $\alpha$ to get

$$\fbox{$\displaystyle \alpha \approx \ensuremath{\frac{\Im{Y^{Rect}_a(0)}\cdot 4\pi}{N^3}}.$}$$ (130)