Eigenfunctions via Integral Equations

To illustrate this integral equation, consider the boundary value problem for several of the familiar orthogonal functions.

1. Trigonometric functions:
   

   $$\frac{d^2u}{dx^2}+\lambda u = 0~~\quad~~ u(0)=u(\ell )=0$$

   $$u(x) = \lambda\int^\ell_0 G(x;\xi )u(\xi )d\xi$$

   $$G(x;\xi) = \frac{1}{\ell}\left\{\begin{array}{ll} x(\ell -\xi ) &\textrm{when}~x<\xi\\ \xi (\ell -x) &\textrm{when}~\xi <x\end{array}\right.$$

   
   Eigenfunction: $u_n(x)=\sin\frac{n\pi x}{\ell}$ ; $\lambda = \left(\frac{n\pi} {\ell}\right)^2$ ; $n=$ integer.
2. Bessel functions:
   

   $$\frac{1}{x}\frac{d}{dx}x\frac{du}{dx}+\left(\lambda -\frac{n^2}{x^2}\right) u = 0~~\quad~~ u~\textrm{finite~at}~x=0\,,\,\infty$$

   $$u(x) = \lambda\int^\infty_0 G(x;\xi ) u(\xi)\xi d\xi$$

   $$G(x;\xi) = \frac{1}{2n}\left\{\begin{array}{ll} \left(\frac{x}{\xi}\right)^n... ...}~x<\xi\\ \left(\frac{\xi}{x}\right)^n &\textrm{when}~\xi <x\end{array}\right.$$

   
   Eigenfunction: $u_n (x) =J_n(\sqrt{\lambda} x)$ ; $0<\lambda <\infty$
3. Legendre polynomials:
   

   $$\frac{d}{dx} (1-x^2)\frac{du}{dx} +\lambda u = 0~~\quad~~u~\textrm{is finite at}~x=\pm 1$$

   $$u(x) = \lambda\int^1_{-1} G(x;\xi )u(\xi )d\xi -\frac{1}{2}\int^1_{-1}u(\xi ) d\xi$$

   $$G(x;\xi) = \frac{1}{2}\left\{\begin{array}{ll} \ln\left(\frac{1+x}{1-\xi}\ri... ...xi\\ \ln\left(\frac{1+\xi}{1-x}\right) &\textrm{when}~\xi <x\end{array}\right.$$

   $$u_n(x) = P_n (x)\,;~~\lambda =n (n +1)\,;~~n =\textrm{integer}\,.$$