In Weissinger's L-Method the circulation G can be expanded into Fourier series after performing the \(y *=\cos \phi_{j}\) transformation. Using the trapezoidal rule for integration the angle of attack at \(j\) th station can be written

\[
\alpha_{j}=\left[2 b_{j j}+\frac{l}{b_{j}} g_{j j}\right] G_{j}+\sum_{\substack{n=1 \\ n eq j}}^{m}\left[\frac{l}{b_{j}}-2 b_{j n}\right] G_{n}
\]

Here,

\(g_{j n}=\frac{-1}{2(M+1)}\left[\frac{L_{j o} f_{n o}+L_{j, M+1} f_{n, M+1}}{2}+\sum_{i=1}^{M} L_{j i} f_{n i}\right]\) and,

\(L_{j i}=L\left(\phi_{j}, \theta_{i}\right)=L\left(y^{*}, \eta^{*}\right)\) and \(f_{n i}=f_{n}\left(\phi_{i}\right)=\frac{2}{m+1} \sum_{k=1}^{m} k \sin k \phi_{n} \cos k \phi_{i}\).

\(\mathrm{M}\) is a parameter involved in the numerical integration process. Using the formulation given here, solve Problem 4.9 with the unswept \(\mathrm{L}\) formula and compare your results.

Write the Aerodynamic Influence Coefficient matrix for the Weissinger L-Method in terms of \(b_{j n}\) and \(g_{j n}\).

For the low aspect ratio wings, show that the integral given below is

\[
\int_{-\beta(x)}^{\beta(x)} \frac{\sqrt{\beta^{2}(x)-\eta^{2}}}{\bar{y}-\eta} d \eta=\pi \bar{y}
\]