## Fourier Series and Orthogonal PolynomialsThis text for undergraduate and graduate students illustrates the fundamental simplicity of the properties of orthogonal functions and their developments in related series. Starting with a definition and explanation of the elements of Fourier series, the text follows with examinations of Legendre polynomials and Bessel functions. Boundary value problems consider Fourier series in conjunction with Laplace's equation in an infinite strip and in a rectangle, with a vibrating string, in three dimensions, in a sphere, and in other circumstances. An overview of Pearson frequency functions is followed by chapters on orthogonal, Jacobi, Hermite, and Laguerre polynomials, and the text concludes with a chapter on convergence. 1941 edition. |

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### Contents

FOURIER SERIES | 1 |

LEGENDRE POLYNOMIALS | 45 |

BESSEL FUNCTIONS | 69 |

BOUNDARY VALUE PROBLEMS | 91 |

DOUBLE SERIES LAPLACE SERIES | 115 |

THE PEARSON FREQUENCY FUNCTIONS | 142 |

ORTHOGONAL POLYNOMIALS | 149 |

JACOBI POLYNOMIALS | 166 |

HERMITE POLYNOMIALS | 176 |

LAGUERRE POLYNOMIALS | 184 |

CONVERGENCE | 191 |

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application approaches zero arbitrary function becomes infinite boundary value problem Chapter VII constant multiple continuous function convergence coordinates corresponding cos2 cosine sum defined definition denominator denoted derivative differential equation discussion equal evaluation expansion expression finite jumps finite number Fourier series func function f(x function of period graph harmonic polynomial Hermite polynomials hypothesis identity independent Integral representation integral values integrand interval irn(x Jacobi polynomials Jn(x JQ(x kx dx kxdx Laguerre Laplace series Laplace's equation Lebesgue left-hand member Legendre polynomials Legendre series linear combination mathematical nomials non-negative normalizing factor notation nth degree nth order obtained orthogonal polynomials paragraph partial sum particular period 2ir pk(x Pn(x poly power series property of orthogonality recurrence formula relation respect right-hand member satisfies sin2 sn(x solution spherical harmonics string substitution theory tion Tn(x trigonometric sum uniform convergence uniformly bounded vanishes weight function yn(x