Lectures on the theory of elliptic functions 椭圆函数论

Prized for its extensive coverage of classical material, this text is also well regarded for its unusual fullness of treatment and its comprehensive discussion of both theory and applications. The author developes the theory of elliptic integrals, beginning with formulas establishing the existence, formation, and treatment of all three types, and concluding with the most general description of these integrals in terms of the Riemann surface. The theories of Legendre, Abel, Jacobi, and Weierstrass are developed individually and correlated with the universal laws of Riemann. The important contributory theorems of Hermite and Liouville are also fully developed. 1910 ed.

CHAPTER I
　PRELIMINARY NOTIONS
　　1. One-valued function. Regular function. Zeros
　　2. Singular points. Pole or infinity
　　3. Essential singular points
　　4. Remark concerning the zeros and the poles .
　　5. The point at infinity
　　6. Convergence of series
　　7. A one-valued function that is regular at all points of the plane is a constant.
　　8. The zeros and the poles of a one-valued function are necessarily isolated
　Rational Functions
　　9-10. Methods (1) of decomposing a rational fraction into its partial frac- tions; (2) of Tepresenting such a fraction as a quotient of two products of linear factors
Principal Analytical Forms of Rational Functions
　　11. First form: Where the poles and the corresponding principal parts are brought into evidence
　　12. Second form: Where the zeros and the infinities are brought into evidence
　Trigonometric Functions
　　13. Integral transcendental functions
　　14. Results established by Cauchy
　　15, 16. The fundamental theorem of algebra extended by Weierstrass to these integral transcendents
　Infinite Products
　　17, 18. Condition of convergence
　　19. The infinite products expressed through infinite series
　　20, 21. The sine-function
　　22. The cot-function
　　23. Development in series
　　The General Trigonometric Functions
　　24. The general trigonometric function expressed as a rational function of the cot-function
　　25. Decomposition into partial fractions
　　26. Expressed as a quotient of linear factors
　Analytic Functions
　　 27. Domain of convergence. Analytic continuation
　　 28. Example of a function which has no definite derivative
　　 29. The function is one-valued in the plane where the canals have been drawn
　　 30. The process may be reversed
　　 31. Algebraic addition-theorems. Definition of an elliptic function Examples
CHAPTER II
　FUNCTIONS WHICH HAVE ALGEBRAIC ADDITION-THEOREMS
　 32. Examples of functions having algebraic addition-theorems
　 33. The addition-theorem stated
　 34. Meray's eliminant equation
　 35. The existence of this equation is universal for the functions considered
　 36. A formula of fundamental importance for the addition-theorems
　 37. The higher derivatives expressed as rational functions of the function and its first derivative
……
CHAPTER III
CHAPTER IV
CHAPTER V
CHAPTER VI
CHAPTER VII
CHAPTER VIII
CHAPTER IX
CHAPTER X
CHAPTER XI
CHAPTER XII
CHAPTER XIII
CHAPTER XIV
CHAPTER XV
CHAPTER XVI
CHAPTER XVII
CHAPTER XVIII
CHAPTER XIX
CHAPTER XX
CHAPTER XXI
Table of Formulas