Laplace, Lamé, and Bessel's Functions by Isaac Todhunter - PDF

About this book :-
" An Elementary Treatise on Laplace's Functions, Lamé's Functions, and Bessel's Functions " by "Isaac Todhunter" is a classical text that systematically introduces three fundamental classes of "special functions" widely used in mathematics and physics. The book begins with "Laplace's functions", exploring Legendre coefficients, single-variable and two-variable cases, and their applications in potential theory and spherical harmonics. Todhunter emphasizes clear derivations, series expansions, and the geometric intuition behind these functions. The text then develops "Lamé's functions", which generalize Laplace’s functions and arise in problems involving ellipsoidal coordinates and boundary-value problems. Todhunter presents analytic methods, transformations, and integral representations to illustrate their properties, making the topic accessible to advanced students and researchers. The final section focuses on "Bessel functions", important in solving differential equations in cylindrical and spherical geometries. Todhunter provides series solutions, recurrence relations, and integral formulas, linking them to physical applications such as vibrations, heat conduction, and wave propagation. Todhunter’s treatise is historically significant as one of the earliest systematic English expositions of these special functions. It remains a valuable reference for understanding the classical development of "function theory", "differential equations", and their applications in mathematical physics. Its clear, unified approach bridges elementary analysis with advanced techniques, providing foundational insights into the behavior and use of special functions.

Book Detail :-
Title: Laplace, Lamé, and Bessel's Functions by Isaac Todhunter - PDF
Publisher: Macmillan
Year: 1875
Pages: 368
Type: PDF
Language: English
ISBN-10 #: 0469127104
ISBN-13 #: 978-0469127104
License: Public Domain Work
Amazon: Amazon

About Author :-
The author Isaac Todhunter (1820–1884) was a British mathematician known for his work in "mathematical analysis" and "special functions". Educated at Cambridge, he focused on making complex topics accessible through clear, structured exposition, bridging theory and applications for students and researchers. Todhunter authored influential texts, including "Laplace, Lamé, and Bessel’s Functions", covering classical "function theory", differential equations, and applied mathematics. His systematic approach and rigorous proofs helped standardize advanced mathematical education in the 19th century, leaving a lasting legacy in both teaching and research within mathematical analysis and applied mathematics.

Book Contents :-
1. Introduction 2. Other Forms of Legendre’s Coefficients 3. Properties of Legendre’s Coefficients 4. The Coefficients Expressed by Definite Integrals 5. Differential Equation Satisfied by Legendre’s Coefficients 6. The Coefficients of the Second Kind 7. Approximate Values of Coefficients of High Orders 8. Associated Functions 9. Continued Fractions 10. Approximate Quadrature 11. Expansion of Functions in Terms of Legendre’s Coefficients 12. Miscellaneous Propositions 13. Laplace’s Coefficients 14. Laplace’s Coefficients: Additional Investigations 15. Laplace’s Functions 16. Expansion of Functions 17. Other Investigations of the Expansion of Functions 18. Expansion of a Function in Sines and Cosines of Multiple Angles 19. Dirichlet’s Investigation 20. Miscellaneous Theorems 21. Special Curvilinear Coordinates 22. General Curvilinear Coordinates 23. Transformation of Laplace’s Principal Equation 24. Transformation of Laplace’s Secondary Equation 25. Physical Applications 26. Lamé’s Functions 27. Separation of the Terms 28. Special Cases 29. Miscellaneous Propositions 30. Definition of Bessel’s Functions 31. Properties of Bessel’s Functions 32. Fourier’s Expression for Bessel’s Functions 33. Large Roots of Fourier’s Equation 34. Expansions in Series of Bessel’s Functions 35. General Theorems on Expansions 36. Miscellaneous Propositions

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