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 New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | MEDIUM | LARGE Below is the OCR-scanned text from this page: 16 . Jacobian Elliptic Functions and Theta Functions L . M . MILNE-TEOMSON Contents Mathematical Properties . . . . . . . . . . . . . . . . . . . . 16.1. Introduction . . . . . . . . . . . . . . . . . . . . . 16.2. Classification of the Twelve Jacobian Elliptic Functions . . 16.3. Relation of the Jacobian Functions to the Copolar Trio . . 16.4. Calculation of the Jacobian Functions by Use of the Arith- metic-Geometric Mean (A.G.M.) . . . . . . . . . . . 16.5. Special Arguments . . . . . . . . . . . . . . . . . . 16.6. Jacobian Functions when m=O or 1 . . . . . . . . . . . 16.7. Principal Terms . . . . . . . . . . . . . . . . . . . 16.8. Change of Argument . . . . . . . . . . . . . . . . . 16.9. Relations Between the Squares of the Functions . . . . . 16.10. Change of Parameter. . . . . . . . . . . . . . . . . . 16.11. Reciprocal Parameter (Jacobi’s Real Transformation) . . . 16.12. Descending Landen Transformation (Gauss’ Transforma- tion) . . . . . . . . . . . . . . . . . . . . . . . 16.13. Approximation in Terms of Circular Functions . . . . . . 16.14. Ascending Landen Transformation . . . . . . . . . . . 16.15. Approximation in Terms of Hyperbolic Functions . . . . 16.16. Derivatives . . . . . . . . . . . . . . . . . . . . . 16.17. Addition Theorems . . . . . . . . . . . . . . . . . . 16.18. Double Arguments . . . . . . . . . . . . . . . . . . 16.19. Half Arguments . . . . . . . . . . . . . . . . . . . 16.20. Jacobi’s I.maginary Transformation . . . . . . . . . . . . 16.21. Complex Arguments . . . . . . . . . . . . . . . . . 16.22. Leading Terms of the Series in Ascending Powers of u . . . 16.23. Series Expansion in Terms of the Nome p . . . . . . . . 16.24. Integrals of the Twelve Jacobian Elliptic Functions . . . . 16.25. Notation for the Integrals of the Squares of the Twelve Jacobian Elliptic Functions . . . . . . . . . . . . . 16.26. Integrals in Terms of the Elliptic Integral of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . 16.27. Theta Functions; Expansions in Terms of the Nome p . . 16.28. Relations Between the Squares of the Theta Functions . . 16.29. Logarithmic Derivatives of the Theta Functions . . . . . 16.30. Logarithms of Theta Functions of Sum and Difference . . 16.31. Jacobi’s Notation for Theta Functions . . . . . . . . . 16.32. Calculation of Jacobi’s Theta Function @(ulm) by Use of the Arithmetic-Geometric Mean . . . . . . . . . . . Page 569 569 570 570 571 571 571 572 572 573 573 573 573 573 573 574 574 574 574 574 574 575 575 575 575 576 576 576 576 576 577 577 577 * University of Arizona . (Prepared under contract with the National Bureau of Standards . ) 567 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.