General Form of Sturm-Liouville System |
The general form of the Sturm-Liouville system is where . |
Special Cases and Orthogonal Polynomials |
Bessel Functions: For , , , , , and , the Sturm-Liouville equation becomes the Bessel's differential equation which is defined on . The solutions of the Bessel's differential equation is called Bessel Functions of the first kind which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.) Legendre Polynomials: For , , , , , and , the Strangleholds equation becomes the Legendre's differential equation which is defined on . The solutions of the Legendre's differential equation with is called Legendre Polynomials which form a complete orthogonal set on the interval . (Further detail; see plots.) Hermite Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Hermite's differential equation which is defined on . The solutions of the Hermite's differential equation with is called Hermite Polynomials which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.) Laguerre Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Laguerre's differential equation which is defined on . The solutions of the Laguerre's differential equation with is called Laguerre Polynomials which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.) Chebyshev Polynomials: For , , , , , and , the Sturm-Liouville equation becomes the Chebyshev's differential equation which is defined on . The solutions of the Chebyshev's differential equation with is called Chebyshev Polynomials which form a complete orthogonal set on the interval with respect to . (Further detail; see plots.) |