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Harris Hancock: Lectures on the theory of elliptic functions

Published $1910$.


Online copies

hosted by archive.org

Contents

CHAPTER 1 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 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.
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.
Infinite products
17.
18.
19. The infinite products expressed through infinite series
20.
21.
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
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. Méray's eliminant equation
35. The existence of this equation is universal for 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
37a.
38.
39. A form of the general integral of Méray's equation
The Discussion Restricted to One-valued Functions
40. All functions which have the property that $\phi(u+v)$ may be rationally expressed through $\phi(u)$, $\phi'(u)$, $\phi(v)$, $\phi'(v)$ are one-valued
41.
42.
43.
44.
45.
46. Example showing that a function $\phi(u)$ may be such that $\phi(u+v)$ is rationally expressible through $\phi(u)$, $\phi'(u)$, $\phi(v)$, $\phi'(v)$ without having an algebraic addition-theorem
Continuation of the Domain in which the Analytic Function $\phi(u)$ has been Defined, with Proofs that its Characteristic Properties are Retained in the Extended Domain
47. Definition of the function in the neighborhood of the origin
48.
49.
50.
51. The other characteristic properties of the function are also retained. The addition-theorem, while limited to a ring-formed region, exists for the whole region of convergence established for $\phi(u)$