Book:Harris Hancock/Lectures on the theory of elliptic functions

Harris Hancock: Lectures on the theory of elliptic functions

Published $1910$.

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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 $\phi(u)$