Chebyshev Polynomial
 Chebyshev Polynomial
 Orthogonal Polynomials Legendre Hermite Laguerre Chebyshev Chebyshev Plot Resources Bibliography

 Home Membership Magazines Forum Search Member Calculators
 Materials Design Processes Units Formulas Math
 Definition In the Sturm-Liouville Boundary Value Problem, there is a special case called Chebyshev's Differential Equation which is defined as: where is a real number. The solutions of this equation are called Chebyshev Functions of degree . If is a non-negative integer, i.e., , the Chebyshev Functions are often referred to as Chebyshev Polynomials .
 Important Properties Rodrigues' Formula: The Chebyshev Polynomials can be expressed by Rodrigues' formula: where Generating Function: The generating function of a Chebyshev Polynomial is: Orthogonality: Chebyshev Polynomials , , form a complete orthogonal set on the interval with respect to the weighting function . It can be shown that: By using this orthogonality, a piecewise continuous function in can be expressed in terms of Chebyshev Polynomials: where This orthogonal series expansion is also known as a Fourier-Chebyshev Series expansion or a Generalized Fourier Series expansion. Even/Odd Functions: Whether a Chebyshev Polynomial is an even or odd function depends on its degree . Based on , •  is an even function, when is even. •  is an odd function, when is odd. Recurrence Relation: A Chebyshev Polynomial at one point can be expressed by neighboring Chebyshev Polynomials at the same point.
Special Results

 where , , are Chebyshev Polynomials of the second kind. A linear combination of Chebyshev Polynomials of the First (Tn) and Second (Un) Kinds forms the general solution of the Chebyshev Equation.