**Chebyshev polynomials** crop up in many areas of calculus, including numerical integration, orthogonal polynomials and spectral methods for partial differential equations. They can also be used for curve fitting (finding a function that models a curve), interpolation and in multiple other areas of numerical analysis.

The general formula for a Chebyshev polynomial, for an integer n ≥ 0, is:

**T _{n}(x) = cos(n cos^{-1}x) **; -1 ≤ x ≤ 1

## Properties

For n ≥ 2 (Smith, 2011):

- T
_{n}(x) is an n th-order polynomial in x. - When n is an even integer, T
_{n}(x) is an even function. - When n is an odd integer, T
_{n}(x) is an odd function. - T
_{n}(x) has n zeros in the open interval (-1, 1). - T
_{n}(x) has n + 1 extrema in the closed interval [-1, 1].

## Chebyshev Polynomials of the First Kind

Some authors refer to Chebyshev polynomials as just the Chebyshev polynomial of the first kind (T_{n})—a polynomial in x of degree n, defined by the relation (Mason & Handscomb, 2002):

**T _{n}(x) = cos nθ when x = cosθ.**

The following table (Culham, 2020) lists the first 12 Chebyshev Polynomials of the first kind, obtained from Rodrigue’s formula:

T_{0}(x) = 1 |

T_{1}(x) = x |

T_{2}(x) = 2x^{2} – 1 |

T_{3}(x) = 4x^{3} – 3x |

T_{4}(x) = 8x^{4} – 8x^{2} + 1 |

T_{5}(x) = 16x^{5} – 20x^{3} + 5x |

T_{6}(x) = 32x^{6} – 48x^{4} + 18x^{2} – 1 |

T_{7}(x) = 64x^{7} – 112x^{5} + 56x^{3} – 7x |

T_{8}(x) = 128x^{8} – 256x^{6} + 160x^{4} – 32x^{2} + 1 |

T_{9}(x) = 256x^{9} – 576x^{7} + 432x^{5} – 120x^{3} + 9x |

T_{10}(x) = 512x^{10} – 1280x^{8} + 1120x^{6} – 400x^{4} + 50x^{2} – 1 |

T_{11}(x) = 1024x^{11} – 2816x^{9} + 2616x^{7} – 1232x^{5} + 220x^{3} – 11x |

## References

Culham, J. (2020). Chebyshev Polynomials. Retrieved August 22, 2020 from: mhtl.uwaterloo.ca/courses/me755/web_chap6.pdf

Mason, J. & Handscomb, S. (2002). Chebyshev Polynomials. CRC Press.

Smith, J.O. Spectral Audio Signal Processing, http://ccrma.stanford.edu/~jos/sasp/, online book, 2011 edition, accessed August 23, 2020.

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