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Ellipsoidal Harmonics. Theory and Applications
George Dassios
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Últimas novedades matemáticas
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The sphere is what might be called a perfect shape. Unfortunately nature is imperfect and many bodies are better represented by an ellipsoid. The theory of ellipsoidal harmonics, originated in the nineteenth century, could only be seriously applied with the kind of computational power available in recent years. This, therefore, is the first book devoted to ellipsoidal harmonics. Topics are drawn from geometry, physics, biosciences and inverse problems. It contains classical results as well as new material, including ellipsoidal bi-harmonic functions, the theory of images in ellipsoidal geometry and vector surface ellipsoidal harmonics, which exhibit an interesting analytical structure. Extended appendices provide everything one needs to solve formally boundary value problems. End-of-chapter problems complement the theory and test the reader’s understanding. The book serves as a comprehensive reference for applied mathematicians, physicists, engineers and for anyone who needs to know the current state of the art in this fascinating subject.
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Prologue 1. The ellipsoidal system and its geometry 2. Differential operators in ellipsoidal geometry 3. Lamé functions 4. Ellipsoidal harmonics 5. The theory of Niven and Cartesian harmonics 6. Integration techniques 7. Boundary value problems in ellipsoidal geometry 8. Connection between sphero-conal and ellipsoidal harmonics 9. The elliptic functions approach 10. Ellipsoidal bi-harmonic functions 11. Vector ellipsoidal harmonics 12. Applications to geometry 13. Applications to physics 14. Applications to low-frequency scattering theory 15. Applications to bioscience 16. Applications to inverse problems Epilogue Appendix A. Background material Appendix B. Elements of dyadic analysis Appendix C. Legendre functions and spherical harmonics Appendix D. The fundamental polyadic integral Appendix E. Forms of the Lamé equation Appendix F. Table of formulae Appendix G. Miscellaneous relations Bibliography Index.
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