Elementary Treatise on Fourier's Series

Name: Elementary Treatise on Fourier's Series | and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical
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and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical

William Elwood Byerly(Autor*in)

Dover Publications (Verlag)

Erschienen am 5. März 2014

498 Seiten

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Inhalt

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. 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.
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. 29. Sine series an odd function of the variable, cosine series an even function, and both series periodic functions.
Art. 30. Development in series involving both sines and cosines of whole multiples of the variable. Fourier's series. Examples.
Art. 31. Extension of the range within which the function and the series are equal. Examples.
Art. 32. Fourier's Integral obtained.
Chapter III. Convergence of Fourier's Series
Arts. 33-36. The question of the convergence of the sine series for unity considered at length.
Arts. 37-38. Statement of the conditions which are sufficient to warrant the development of a function into a Fourier's series. Historical note.
Art. 39. Graphical representation of successive approximations to a sine series. Properties of a Fourier's series inferred from the constructions.
Arts. 40-42. Investigation of the conditions under which a Fourier's series can be differentiated term by term.
Art. 43. Conditions under which a function can be expressed as a Fourier's Integral.
Chapter IV. Solution of Problems in Physics by the Aid of Fourier's Integrals and Fourier's Series
Arts. 44-48. Logarithmic Potential. Flow of electricity in an infinite plane, where the value of the Potential Function is given along an infinite straight line
along two mutually perpendicular straight lines
along two parallel straight lines. Examples. Use of Conjugate Functions. Sources and Sinks. Equipotential lines and lines of Flow. Examples.
Arts. 49-52. One-dimensional flow of heat. Flow of heat in an infinite solid
in a solid with one plane face at the temperature zero
in a solid with one plane face whose temperature is a function of the time (Riemann's solution)
in a bar of small cross section from whose surface heat escapes into air at temperature zero. Limiting state approached when the temperature of the origin is a periodic function of the time. Examples.
Arts. 53-54. Temperatures due to instantaneous and to permanent heat sources and sinks, and to heat doublets. Examples. Application to the case where there is leakage.
Arts. 55-56. Transmission of a disturbance along an infinite stretched elastic string. Examples.
Arts. 57-58. Stationary temperatures in a long rectangular plate. Temperature of the base unity. Summation of a Trigonometric series. Isothermal lines and lines of flow. Examples.
Art. 59. Potential Function given along the perimeter of a rectangle. Examples.
Arts. 60-63. One-dimensional flow of heat in a slab with parallel plane faces. Both faces at temperature zero. Both faces adiathermanous. Temperature of one face a function of the time. Examples.
Art. 64. Motion of a stretched elastic string fastened at the ends. Steady vibration. Nodes. Examples.
Art. 65. Motion of a string in a resisting medium.
Art. 66. Flow of heat in a sphere whose surface is kept at a constant temperature.
Arts. 67-68. Cooling of a sphere in air. Surface condition given by a differential equation. Development in a Trigonometric series of which Fourier's Sine Series is a special case. Examples.
Arts. 69-70. Flow of heat in an infinite solid with one plane face which is exposed to air whose temperature is a function of the time. Solution for an instantaneous heat source when the temperature of the air is zero. Examples.
Arts. 71-73. Vibration of a rectangular drumhead. Development of a function of two variables in a double Fourier's Series. Examples. Nodal lines in a rectangular drumhead. Nodal lines in a square drumhead.
Miscellaneous Problems
I. Logarithmic Potential. Polar Coordinates
II. Potential Function in Space
III. Conduction of heat in a plane
IV. Conduction of heat in Space
Chapter V. Zonal Harmonics
Art. 74. Recapitulation. Surface Zonal Harmonics (Legendrians). Zonal Harmonics of the second kind.
Arts. 75-76. Legendrians as coefficients in a Power Series. Special values.
Art. 77. Summary of the properties of a Legendrian. List of the first eight Legendrians. Relation connecting any three successive Legendrians.
Arts. 78-81. Problems in Potential. Potential Function due to the attraction of a material circular ring of small cross section. Potential Function due to a charge of electricity placed on a thin circular disc. Examples: Spheroidal conductors. Potential Function due to the attraction of a material homogeneous circular disc. Examples : Homogeneous hemisphere
Heterogeneous sphere
Homogeneous spheroids. Generalisation.
Art. 82. Legendrian as a sum of cosines.
Arts. 83-84. Legendrian as the mth derivative of the mth power
Art. 85. Equations derivable from Legendre's Equation.
Art. 86. Legendrian as a Partial Derivative.
Art. 87. Legendrian as a Definite Integral.
Arts. 88-90. Development in Zonal Harmonic Series. Integral of the product of two Legendrians of different degrees. Integral of the square of a Legendrian. Formulas for the coefficients of the series.
Arts. 91-92. Integral of the product of two Legendrians obtained by the aid of Legendre's Equation
by the aid of Green's Theorem. Additional formulas for integration. Examples.
Arts. 93-94. Problems in Potential where the value of the Potential Function is given on a spherical surface and has circular symmetry about a diameter. Examples.
Art. 95. Development of a power of x in Zonal Harmonic Series.
Art. 96. Useful formulas.
Art. 97. Development of sin n? and cos n? in Zonal Harmonic Series. Examples. Graphical representation of the first seven Surface Zonal Harmonics. Construction of successive approximations to Zonal Harmonic Series.
Arts. 98-99. Method of dealing with problems in Potential when the density is given. Examples.
Art. 100. Surface Zonal Harmonics of the second kind. Examples: Conal Harmonics.
Chapter VI. Spherical Harmonics
Arts. 101-102. PArticular Solutions of Laplace's Equation obtained. Associated Functions. Tesseral Harmonics. Surface Spherical Harmonics. Solid Spherical Harmonics. Table of Associated Functions. Examples.
Arts. 103-108. Development in Spherical Harmonic Series. The integral of the product of two Surface Spherical Harmonics of different degrees taken over the surface of the unit sphere is zero. Examples. The integral of the product of two Associated Functions of the same order. Formulas for the coefficients of the series. Illustrative example. Examples.
Arts. 109-110. Any homogeneous rational integral Algebraic function of x, y, and z which satisfies Laplace's Equation is a Solid Spherical Harmonic. Examples.
Art. 111. A transformation of axes to a new set having the same origin will change a Surface Spherical Harmonic into another of the same degree.
Arts. 112-114. Laplacians. Integral of the product of a Surface Spherical Harmonic by a Laplacian of the same degree. Development in Spherical Harmonic Series by the aid of Laplacians. Table of Laplacians. Example.
Art. 115. Solution of problems in Potential by direct integration. Examples.
Arts. 116-118. Differentiation along an axis. Axes of a Spherical Harmonic.
Art. 119. Roots of a Zonal Harmonic. Roots of a Tesseral Harmonic. Nomenclature justified.
Chapter VII. Cylindrical Harmonics (Bessel's Functions)
Art. 120. Recapitulation. Cylindrical Harmonics (Bessel's Functions) of the zeroth order
of the nth order
of the second kind. General solution of Bessel's Equation.
Art. 121. Bessel's Functions as definite integrals. Examples.
Art. 122. Properties of Bessel's Functions. Semi-convergent series for a Bessel's Function. Examples.
Art. 123. Problem: Stationary temperatures in a cylinder (a) when the temperature of the convex surface is zero
(b) when the convex surface is adiathermanous
(c) when the convex surface is exposed to air at the temperature zero.
Art. 124. Roots of Bessel's functions.
Art. 125. The integral of r times the product of two Cylindrical Harmonics of the zeroth order. Example.
Art. 126. Development in Cylindrical Harmonic Series. Formulas for the coefficients. Examples.
Art. 127. Problem: Stationary temperatures in a cylindrical shell. Bessel's Functions of the second kind employed. Example: Vibration of a ring membrane.
Art. 128. Problem: Stationary temperatures in a cylinder when the temperature of the convex surface varies with the distance from the base. Bessel's Functions of a complex variable. Examples.
Art. 129. Problem: Stationary temperatures in a cylinder when the temperatures of the base are unsymmetrical. Bessel's Functions of the nth order employed. Miscellaneous examples. Bessel's Functions of fractional order.
Chapter VIII. Laplace's Equation in Curvilinear Coordinates. Ellipsoidal Harmonics
Arts. 130-131. Orthogonal Curvilinear Coordinates in general. Laplace's Equation expressed in terms of orthogonal curvilinear co ördinates by the aid of Greein's theorem.
Arts, 132-135. Spheroidal Coordinates. Laplace's Equation in spheroidal coordinates, in normal spheroidal coördinates. Examples. Condition that a set of curvilinear coördinates should be normal. Thermometric Parameters. Particular solutions of Laplace's Equation in spheroidal coordinates. Spheroidal Harmonics. Examples. The Potential Function due to the attraction of an oblate spheroid. Solution for an external point. Examples.
Arts. 136-141. Ellipsoidal Coördinates. Laplace's Equation in ellipsoidal coördinates. Normal ellipsoidal coordinates expressed as Elliptic Integrals. Particular solutions of Laplace's Equation. Lamé's Equation. Ellipsoidal Harmonics (Lamé's Functions). Tables of Ellipsoidal Harmonics of the degrees 1, 2, and 3. Lamp's Functions of the second kind. Examples. Development in Ellipsoidal Harmonic series. Value of the Potential Function at any point in space when its value is given at all points on
Art. 142. Conical Coördinates. The product of two Ellipsoidal Harmonics a Spherical Harinonic.
Art. 143. Toroidal Coördinates. Laplace's Equation ih toroidal coordinates. Particular solutions. Toroidal Harmonics. Potential Function for an anchor ring.
Chapter IX. Historical Summary
Appendix
Tables
Table I. Surface Zonal Harmonics. Argument ?
Table II. Surface Zonal Harmonics. Argument x
Table III. Hyperbolic Functions
Table IV. Roots of Bessel's Functions
Table V. Roots of Bessel's Functions
Table VI. Bessel's Functions

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