An Elementary Treatise on Fouriers SeriesAn Elementary Treatise on Fourier's Series

Originally published over a century ago, this work remains among the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics. The subsequent growth of science into a diverse range of specialties has enhanced the value of this classic, whose thorough, basic treatment presents material that is assumed in many other studies but seldom available in such concise form. The development of functions, series, and their differential equations receives detailed explanations, and throughout the text, theory is applied to practical problems, with the solutions fully worked out. In addition, 190 problems, many with hints, are included. 1893 edition. Appendix of 6 tables. Originally published over a century ago, this work remains among the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics. The subsequent growth of science into a diverse range of specialties has enhanced the value of this classic, whose thorough, basic treatment presents material that is assumed in many other studies but seldom made available in such concise form. The development of functions, series, and their differential equations receives concrete and detailed explanations. Throughout the text, theory is applied to practical problems, with the solutions fully and clearly worked out. In addition, readers are presented with 190 problems to solve, many appearing with helpful hints as to method. An appendix includes tables of surface zonal harmonics, hyperbolic functions, and Bessel's function. As a handbook, reference, and lucid exposition of fundamental methods, there is no substitute for this book in such areas as mathematical physics, wave mechanics, and advanced engineering, including communications technology, aerodynamics, radar, acoustics, and other fields. Cover Page; Title Page; Copyright Page; Preface; Contents; Chapter I. Introduction; Art. 1. List of some important homogeneous linear partial differential equations of Physics.; Arts. 2-4. Distinction between the general solution and a particular solution of a differential equation. Need of additional data to make the solution of a differential equation determinate. Definition of linear and of linear and-homogeneous.; Arts. 5-6. Particular solutions of homogeneous linear differential equations may be combined into a more general solution. Need of development in terms of normal forms Art. 16. Application to Legendre's Equation. Several forms of general solution obtained. Zonal Harmonics of the second kind. Art. 17. Application to Bessel's Equation. General solution obtained for the case where m is not an integer, and for the case where m is zero. Bessel's Function of the second kind and zeroth order.; Art. 18. Method of obtaining the general solution of an ordinary linear differential equation of the second order from a given particular solution. Application to the equations considered in Arts. 14-17.; Chapter II. Development In Trigonometric Series Arts. 19-22. Determination of the coefficients of n terms of a sine series so that the sum of the terms shall be equal to a given function of x for n given values of x. Numerical example. Art. 23. Problem Of development in sine series treated as a limiting case of the problem just solved.; Arts. 24-25. Shorter method of solving the problem of development in series involving sines of whole multiples of the variable. Working rule deduced. Recapitulation.; Art. 26. A few important sine developments obtained. Examples.; Arts. 27-28. Development in cosine series. Examples Art. 7. Problem: Permanent state of temperatures jn a thin rectangular plate. Need of a development in sine series. Example. Art. 8. Problem: Transverse vibrations of a stretched elastic string. A development in sine series suggested.; Art. 9. Problem: Potential function due to the attraction of a circular ring of small cross-section. Surface Zonal Harmonics (Legendre's Coefficients). Example.; Art. 10. Problem: Permanent state of temperatures in a solid sphere. Development in terms of Surface Zonal Harmonics suggested Arts. 11-12. Problem: Vibrations of a circular drumhead. Cylindrical Harmonics (Bessel's Functions). Recapitulation. Art. 13. Method of making the solution of a linear partial differential equation depend upon solving a set of ordinary differential equations by assuming the dependent variable equal to a product of factors each of which involves but one of the independent variables.; Arts. 14-15. Method of solving ordinary homogeneous linear differential equations by development in power series. Applications

One of the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics, this classic offers a basic but thorough treatment of material that is assumed in many other studies but rarely available in concise form. Includes 190 problems, approximately half with answers. 1893 edition.

Art. 29. Sine series an odd function of the variable, cosine series an even function, and both series periodic functions