## An Elementary Treatise on Fourier's Series and Spherical, Cylindric, and Ellipsoidal Harmonics: With Applications to Problems in Mathematical PhysicsFirst published in 1893, Byerly's classic treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics has been used in classrooms for well over a century. This practical exposition acts as a primer for fields such as wave mechanics, advanced engineering, and mathematical physics. Topics covered include: . development in trigonometric series . convergence on Fourier's series . solution of problems in physics by the aid of Fourier's integrals and Fourier's series . zonal harmonics . spherical harmonics . cylindrical harmonics (Bessel's functions) . and more. Containing 190 exercises and a helpful appendix, this reissue of Fourier's Series will be welcomed by students of higher mathematics everywhere. American mathematician WILLIAM ELWOOD BYERLY (1849-1935) also wrote Elements of Differential Calculus (1879) and Elements of Integral Calculus (1881). |

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### Contents

1 | |

11 | |

Somjtioh of Problems i Physics sr the Aid op Fouriers Integrals | 30 |

Development is Trigonometric Series 3064 | 48 |

Arts 4448 Logarithmic Potential Flow of electricity in an infinite plane | 53 |

4 Temperatures due to instantaneous and to permanent heat sources and sinks | 60 |

Arts 8890 Daetopment in Zonal Harmonic Series Integral of the product of | 174 |

CHAPTER VI | 195 |

CHAPTER VII | 219 |

CHAPTER IX | 235 |

Historical Srauumr 267276 | 267 |

Tables 277287 | 277 |

Laplaces Equation is Cvbvimhsar Coordinates Ellipsoidal Harmonics 588286 | 286 |

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### Common terms and phrases

abscissas approaches axis Bessel's Functions coefficients convergent corresponding cosh cosine cosine series Crelle's Journal curve curvilinear coordinates cylinder differential equation ease ellipsoid Ellipsoidal Harmonics equal to zero equipotential EXAMPLES expressed external point finite discontinuities flow of heat formula Fourier's Series given function Green's Theorem Hence homogeneous indefinitely increased infinite initially distorted involving kmAx Lamp's Laplace's Equation Legendre's Equation limiting value membrane method mth degree multiply mx.dx obtained odd function parallelepiped particular solution plane Pm(x positive integer potential function potential function due problem radius rectangular reduces required solution roots satisfy second member Show sin x sinh slab sm g solid solution of Laplace's solution of Legendre's solve Spherical Harmonic string substitute Surface Spherical Harmonic symmetrical temperature zero theory Trigonometric series variable vibration whence whole number Zonal Harmonics

### Popular passages

Page 19 - Let these roots be called a and /S, then is a solution, and since it contains two arbitrary constants it is the general solution.

Page 12 - ... about an axis through the centre perpendicular to the plane of the figure.

Page 9 - As (5) must be true no matter what the value of x, the coefficient of any given power of x, as for instance a;*, must vanish. Hence (k + 2) (A + 1K+, - k(k + l)at + m(m + !)лл— 0 (6) m...