Elliptic Functions: A Primer defines and describes what is an elliptic function, attempts to have a more elementary approach to them, and drastically reduce the complications of its classic formulae; from which the book proceeds to a more detailed study of the subject while being reasonably complete in itself. The book squarely faces the situation and acknowledges the history of the subject through the use of twelve allied functions instead of the three Jacobian functions and includes its applications for double periodicity, lattices, multiples and sub-multiple periods, as well as many others in trigonometry. Aimed especially towards but not limited to young mathematicians and undergraduates alike, the text intends to have its readers acquainted on elliptic functions, pass on to a study in Jacobian elliptic functions, and bring a theory of the complex plane back to popularity.
Inhalt
Editor's Preface
List of Tables
Chapter 1. Double periodicity
Equivalent bases
Chapter 2. Lattices
Chapter 3. Multiples and sub-multiples of periods
Chapter 4. Fundamental parallelogram
Liouville's theorem-a doubly periodic function without accessible singularities is a constant
Chapter 5. Definition of an elliptic function
A rational function of an elliptic function is an elliptic function
Chapter 6. An elliptic function (unless constant) has poles and zeros
Identification of an elliptic function
(i) by poles and principal parts
(ii) by poles and zeros
Chapter 7. Residue sum of an elliptic function is zero
Chapter 8. Derivative of an elliptic function
Order of an elliptic function
No functions of the first order
Chapter 9. Additive pseudoperiodicity
Integration of an elliptic function with zero residues
Signature
Evaluation of Aß - Ba for a function additively pseudoperiodic in a, ß with moduli A, B
Chapter 10. Pole-sum of an elliptic function
Chapter 11. The mid-lattice points
Odd and even elliptic functions
Chapter 12. Construction of the function z
Chapter 13. Construction and periodicity of the Weierstrassian function Pz
Chapter 14. Zeros of P'z
The constants ef, eg, eh
Construction of the primitive functions fj z, gj z, hj z
Chapter 15. Periodicity of the primitive functions
Primitive functions are odd functions with simple poles
Structure patterns and residue patterns
Double series for fj z
Chapter 16. Construction and pseudoperiodicity of z
The constants f, g, h
Laurent series for z, Oz, 2k-1z, 2kz
Chapter 17. Construction of sz
Chapter 18. Construction, in terms of z and Pz of an elliptic function with assigned poles and principal parts
Expression for...
Constant value of...
Chapter 19. Construction, in terms of óæ, of an elliptic function with assigned poles and zeros
Expression for...
Expression for the primitive function pjz
Chapter 20. Expression of an elliptic function in the form ...
Chapter 21. Expression for.'2z in terms of.z
Evaluation of...
Chapter 22. Expression of an elliptic function in the form S ...
Chapter 23. Elliptic functions on the same lattice are connected algebraically
Chapter 24. The six critical constants pq
f2g + g2g + h2f =0
fgfh = gfhf
gr = vfg
Chapter 25. Quarter-period addition to the argument of a primitive function
The twelve elementary functions
pq z qp z = qp'wq; pqz qrz = pqwr, prz
Periods and poles of pq z
Relations between the squares of the elementary functions
Chapter 26. The functions pz and pqz as solutions of differential equations
Chapter 27. Copolar functions and simultaneous differential equations
Chapter 28. Addition theorems for pz and .z and .z
...+ fj'z/fjz
Chapter 29. Addition theorems for fjz, jfz and hgz
Chapter 30. Symmetrical algebraic relations between fjx, fjy, fjz, x + z = 0
Chapter 31. Integration of rational functions of .z and .'z
Integration of functions rational in the primitive functions
Chapter 32. The functions .z and pqz as inverted integrals
Chapter 33. Statements of the inversion theorem
Chapter 34. The Weierstrassian half-periods as definite integrals
Chapter 35. Standardisation of an elliptic integral
The normalising factor and the Jacobian lattice
Chapter 36. Definition of the Jacobian functions
Chapter 37. Periodicity of pqu
Solution of pqu = ±pqa
Chapter 38. Parameters and moduli
The constant pqKr
Chapter 39. Leading coefficients
Linear relations between squares of copolar Jacobian functions
Quarter-period addition
Chapter 40. Derivatives and differential equations
Chapter 41. The Jacobian functions as inverted integrals
Chapter 42. The Jacobian quarter-periods as definite integrals
The functions X(c), X'(c)
The ranges of the twelve Jacobian functions for 0 = c < 1 ; 0 = u = X
Chapter 43. Addition theorems for the Jacobian functions
Chapter 44. Jacobi's imaginary and real transformations
Chapter 45. Duplication
The bipolar function bpqu
ps 2u + qs 2u = brsu
2 ps 2u = bqsu + brsu - bpsu
ps2u = ps2Kr (1 + qp 2u)/(1 - rp 2u)
Chapter 46. The Landen transformations
Chapter 47. The reduction of a rational function of Jacobian functions
Chapter 48. Integration of the Jacobian function pqu
Integration of functions of the form pqu ø(pq2u)
Chapter 49. The integrating function Pqu
Linear relations between integrating functions
Pseudoperiodicity of the integrating functions
The half-moduli Nn, Cc
Legendre's identity
Interchange of Sew and Snw under Jacobi's imaginary transformation
The constants K, K', E, E'
Addition theorems for integrating functions
Chapter 50. Integration of a polynomial in the squares of Jacobian functions
Chapter 51. The function IIs (u, a)
Relation of IIs (u, a) and óu
Chapter 52. Differentiation of Jacobian functions
Integrating functions with respect to the parameter c
Chapter 53. Degeneration of Jacobian systems (c = 0) to circular functions
Degeneration of Jacobian systems (c = 1) to hyperbolic functions
First approximations to functions with a small parameter
Chapter 54. The c-derivatives of Kc, Kn, DsKc, DsKn
The quarter-period differential equation and its solution
X'...
X' = ...
2...
f...
Legendre's identity
Chapter 55. Differentiation of Weierstrassian functions with respect to h2, h3
Chapter 56. Weierstrassian and elementary functions with an axial basis
Distribution of real values of...
Variation of .z and pq z on the perimeter of the basic rectangle JFHG<…