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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.)
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