| 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 |
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