Exercise 4.4.1

Let $h_k=\frac{\mathit{j\pi kh}}{N+1}$, pick the $k$th row in equation (3), we have

$\sin <!--\; FUNCTION\; APPLICATION\; -->h_{k-1}-2\sin <!--\; FUNCTION\; APPLICATION\; -->h_k+\sin <!--\; FUNCTION\; APPLICATION\; -->h_{k+1}={\lambda }_j\sin <!--\; FUNCTION\; APPLICATION\; -->h_k$, solve for ${\lambda }_j$, we have

${\lambda }_j=\frac{\sin <!--\; FUNCTION\; APPLICATION\; -->h_{k-1}+\sin <!--\; FUNCTION\; APPLICATION\; -->h_{k+1}}{\sin <!--\; FUNCTION\; APPLICATION\; -->h_k}-2=\frac{2\sin <!--\; FUNCTION\; APPLICATION\; -->\frac{h_{k-1}+h_{k+1}}{2}\cos <!--\; FUNCTION\; APPLICATION\; -->\frac{h_{k-1}-h_{k+1}}{2}}{\sin <!--\; FUNCTION\; APPLICATION\; -->h_k}-2=\frac{2\sin <!--\; FUNCTION\; APPLICATION\; -->h_k\cos <!--\; FUNCTION\; APPLICATION\; -->h_1}{\sin <!--\; FUNCTION\; APPLICATION\; -->h_k}-2=2(\cos <!--\; FUNCTION\; APPLICATION\; -->h_1-1)=2(\cos <!--\; FUNCTION\; APPLICATION\; -->\frac{\mathit{j\pi h}}{N+1}-1)$