| CHAPTER 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 approxima tion 15. Least-square property 16. Parseval's theorem 17. Summation of series 18. Fejer's theorem for a continuous function 19. Proof of Weierstrass's theorem by means of de la Vall~e 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 CHAPTER 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 Yo(x) 3. Orthogonality 4. Integral representation of Jo(x) 5. Zeros of Yo(x) and related functions 6. Expansion of an arbitrary function in series 7. Definition of J.(x) 8. Orthogonality: developments in series 9. Integral representation of J.(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; Pois- son's integral 7. Transformation of Laplace's equation in three di- mensions 8. Legendre series: Laplace's equation in a sphere CHAPTER V.DOUBLE Series;LAPLACE sERIES VI.THE PEARSON FREQUENCY FUNCTIONS VII.ORTHOGONAL POLYNOMIALS CHAPTER IX.HERMITE POLYNOMIALS X.LAGUERRE POLYNOMIALS XI.CONVERGENCE EXERCISES BIBLIOGRAPHY INDEX |
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