目录
 前辅文
 I. FOURIER SERIES
  1. Definition of Fourier series
  2. Orthogonality of sines and cosines
  3. Determination of the coefficients
  4. Series of cosines and series of sines
  5. Examples
  6. Magnitude of coefficients under special hypotheses
  7. Riemann's theorem on limit of general coefficient
  8. Evaluation of a sum of cosines
  9. Integral formula for partial sum of Fourier series
  10. Convergence at a point of continuity
  11. Uniform convergence under special hypotheses
  12. Convergence at a point of discontinuity
  13. Sufficiency of conditions relating to a restricted neighborhood
  14. Weierstrass's theorem on trigonometric approximation
  15. Least-square property
  16. Parseval's theorem
  17. Summation of series
  18. Fejér's theorem for a continuous function
  19. Proof of Weierstrass's theorem by means of de la Vallée Poussin's integral
  20. The Lebesgue constants
  21. Proof of uniform convergence by the method of Lebesgue
 II. LEGENDRE POLYNOMIALS
  1. Preliminary orientation
  2. Definition of the Legendre polynomials by means of the generating function
  3. Recurrence formula
  4. Differential equation and related formulas
  5. Orthogonality
  6. Normalizing factor
  7. Expansion of an arbitrary function in series
  8. Christoffel's identity
  9. Solution of the differential equation
  10. Rodrigues's formula
  11. Integral representation
  12. Bounds of Pn(x)
  13. Convergence at a point of continuity interior to the interval
  14. Convergence at a point of discontinuity interior to the interval
 III. BESSEL FUNCTIONS
  1. Preliminary orientation
  2. Definition of J0(x)
  3. Orthogonality
  4. Integral representation of J0(x)
  5. Zeros of J0(x) and related functions
  6. Expansion of an arbitrary function in series
  7. Definition of Jn(x)
  8. Orthogonality: developments in series
  9. Integral representation of Jn(x)
  10. Recurrence formulas
  11. Zeros
  12. Asymptotic formula
  13. Orthogonal functions arising from linear boundary value problems
 IV. BOUNDARY VALUE PROBLEMS
  1. Fourier series: Laplace's equation in an infinite strip
  2. Fourier series: Laplace's equation in a rectangle
  3. Fourier series: vibrating string
  4. Fourier series: damped vibrating string
  5. Polar coordinates in the plane
  6. Fourier series: Laplace's equation in a circle; Poisson's integral
  7. Transformation of Laplace's equation in three dimensions
  8. Legendre series: Laplace's equation in a sphere
  9. Bessel series: Laplace's equation in a cylinder
  10. Bessel series: circular drumhead
 V. DOUBLE SERIES; LAPLACE SERIES
  1. Boundary value problem in a cube; double Fourier series
  2. General spherical harmonics
  3. Laplace series
  4. Harmonic polynomials
  5. Rotation of axes
  6. Integral representation for group of terms in the Laplace series
  7. Completeness of the Laplace series
  8. Boundary value problem in a cylinder; series involving Bessel functions of positive order
 VI. THE PEARSON FREQUENCY FUNCTIONS
  1. The Pearson differential equation
  2. Quadratic denominator, real roots
  3. Quadratic denominator, complex roots
  4. Linear or constant denominator
  5. Finiteness of moments
 VII. ORTHOGONAL POLYNOMIALS
  1. Weight function
  2. Schmidt's process
  3. Orthogonal polynomials corresponding to an arbitrary weight function
  4. Development of an arbitrary function in series
  5. Formula of recurrence
  6. Christoffel-Darboux identity
  7. Symmetry
  8. Zeros
  9. Least-square property
  10. Differential equation
 VIII. JACOBI POLYNOMIALS
  1. Derivative definition
  2. Orthogonality
  3. Leading coefficients
  4. Normalizing factor; series of Jacobi polynomials
  5. Recurrence formula
  6. Differential equation
 IX. HERMITE POLYNOMIALS
  1. Derivative definition
  2. Orthogonality and normalizing factor
  3. Hermite and Gram-Charlier series
  4. Recurrence formulas; differential equation
  5. Generating function
  6. Wave equation of the linear oscillator
 X. LAGUERRE POLYNOMIALS
  1. Derivative definition
  2. Orthogonality; normalizing factor; Laguerre series
  3. Differential equation and recurrence formulas
  4. Generating function
  5. Wave equation of the hydrogen atom
 XI. CONVERGENCE
  1. Scope of the discussion
  2. Magnitude of the coefficients; first hypothesis
  3. Convergence; first hypothesis
  4. Magnitude of the coefficients; second hypothesis
  5. Convergence; second hypothesis
  6. Special Jacobi polynomials
  7. Multiplication or division of the weight function by a polynomial
  8. Korous's theorem on bounds of orthonormal polynomials
 EXERCISES
 BIBLIOGRAPHY
 INDEX
 前辅文
 I. FOURIER SERIES
  1. Definition of Fourier series
  2. Orthogonality of sines and cosines
  3. Determination of the coefficients
  4. Series of cosines and series of sines
  5. Examples
  6. Magnitude of coefficients under special hypotheses
  7. Riemann's theorem on limit of general coefficient
  8. Evaluation of a sum of cosines
  9. Integral formula for partial sum of Fourier series
  10. Convergence at a point of continuity
  11. Uniform convergence under special hypotheses
  12. Convergence at a point of discontinuity
  13. Sufficiency of conditions relating to a restricted neighborhood
  14. Weierstrass's theorem on trigonometric approximation
  15. Least-square property
  16. Parseval's theorem
  17. Summation of series
  18. Fejér's theorem for a continuous function
  19. Proof of Weierstrass's theorem by means of de la Vallée Poussin's integral
  20. The Lebesgue constants
  21. Proof of uniform convergence by the method of Lebesgue
 II. LEGENDRE POLYNOMIALS
  1. Preliminary orientation
  2. Definition of the Legendre polynomials by means of the generating function
  3. Recurrence formula
  4. Differential equation and related formulas
  5. Orthogonality
  6. Normalizing factor
  7. Expansion of an arbitrary function in series
  8. Christoffel's identity
  9. Solution of the differential equation
  10. Rodrigues's formula
  11. Integral representation
  12. Bounds of Pn(x)
  13. Convergence at a point of continuity interior to the interval
  14. Convergence at a point of discontinuity interior to the interval
 III. BESSEL FUNCTIONS
  1. Preliminary orientation
  2. Definition of J0(x)
  3. Orthogonality
  4. Integral representation of J0(x)
  5. Zeros of J0(x) and related functions
  6. Expansion of an arbitrary function in series
  7. Definition of Jn(x)
  8. Orthogonality: developments in series
  9. Integral representation of Jn(x)
  10. Recurrence formulas
  11. Zeros
  12. Asymptotic formula
  13. Orthogonal functions arising from linear boundary value problems
 IV. BOUNDARY VALUE PROBLEMS
  1. Fourier series: Laplace's equation in an infinite strip
  2. Fourier series: Laplace's equation in a rectangle
  3. Fourier series: vibrating string
  4. Fourier series: damped vibrating string
  5. Polar coordinates in the plane
  6. Fourier series: Laplace's equation in a circle; Poisson's integral
  7. Transformation of Laplace's equation in three dimensions
  8. Legendre series: Laplace's equation in a sphere
  9. Bessel series: Laplace's equation in a cylinder
  10. Bessel series: circular drumhead
 V. DOUBLE SERIES; LAPLACE SERIES
  1. Boundary value problem in a cube; double Fourier series
  2. General spherical harmonics
  3. Laplace series
  4. Harmonic polynomials
  5. Rotation of axes
  6. Integral representation for group of terms in the Laplace series
  7. Completeness of the Laplace series
  8. Boundary value problem in a cylinder; series involving Bessel functions of positive order
 VI. THE PEARSON FREQUENCY FUNCTIONS
  1. The Pearson differential equation
  2. Quadratic denominator, real roots
  3. Quadratic denominator, complex roots
  4. Linear or constant denominator
  5. Finiteness of moments
 VII. ORTHOGONAL POLYNOMIALS
  1. Weight function
  2. Schmidt's process
  3. Orthogonal polynomials corresponding to an arbitrary weight function
  4. Development of an arbitrary function in series
  5. Formula of recurrence
  6. Christoffel-Darboux identity
  7. Symmetry
  8. Zeros
  9. Least-square property
  10. Differential equation
 VIII. JACOBI POLYNOMIALS
  1. Derivative definition
  2. Orthogonality
  3. Leading coefficients
  4. Normalizing factor; series of Jacobi polynomials
  5. Recurrence formula
  6. Differential equation
 IX. HERMITE POLYNOMIALS
  1. Derivative definition
  2. Orthogonality and normalizing factor
  3. Hermite and Gram-Charlier series
  4. Recurrence formulas; differential equation
  5. Generating function
  6. Wave equation of the linear oscillator
 X. LAGUERRE POLYNOMIALS
  1. Derivative definition
  2. Orthogonality; normalizing factor; Laguerre series
  3. Differential equation and recurrence formulas
  4. Generating function
  5. Wave equation of the hydrogen atom
 XI. CONVERGENCE
  1. Scope of the discussion
  2. Magnitude of the coefficients; first hypothesis
  3. Convergence; first hypothesis
  4. Magnitude of the coefficients; second hypothesis
  5. Convergence; second hypothesis
  6. Special Jacobi polynomials
  7. Multiplication or division of the weight function by a polynomial
  8. Korous's theorem on bounds of orthonormal polynomials
 EXERCISES
 BIBLIOGRAPHY
 INDEX