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        "description": "The target is the 1829 elliptic-function representation framework, with nondegenerate modulus and well-defined elliptic integrals. It is not all prior addition formulas, the later general theory of Abelian functions or rational functions of the complex argument. The account follows the selected translation and original Latin source. A reconstruction of all transformation proofs, exact constants, four-square arithmetic consequences and modern global analytic certification is outside its scope.",
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      "statement": "# Jacobi: elliptic functions, double periods and theta representation\n\nJacobi organized elliptic functions through the inverse amplitude of elliptic integrals, imaginary-argument transformations and two independent periods. His H and theta functions express the resulting elliptic functions as quotients and connect their product and trigonometric-series expansions.\n\nChanging viewpoint from an integral to its inverse makes periodic structure accessible. The complementary modulus connects real and imaginary arguments; H and theta supply numerator and denominator expressions with different shift laws. The same book develops these representations alongside transformations. Legendre and Abel are explicitly acknowledged, and the 1827 announcements and 1829 exposition belong to one contribution family.\n\n## Scientific scope\n\nThe target is the 1829 elliptic-function representation framework, with nondegenerate modulus and well-defined elliptic integrals. It is not all prior addition formulas, the later general theory of Abelian functions or rational functions of the complex argument. The account follows the selected translation and original Latin source. A reconstruction of all transformation proofs, exact constants, four-square arithmetic consequences and modern global analytic certification is outside its scope.\n\n## People and systems\n\nCarl Gustav Jacob Jacobi\n\n\n\n## Original sources\n\n- [C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum (1829), Alexander Aycock English translation, Euler-Kreis Mainz](https://download.uni-mainz.de/mathematik/Algebraische%20Geometrie/Euler-Kreis%20Mainz/Jacobi_Fundamenta%20Nova.pdf)\n- [C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti: Bornträger,1829; ETH Rar5081 original Latin OCR](https://www.e-rara.ch/zut/download/ftpack/plain/1234083)\n",
      "summary": "Jacobi organized elliptic functions through the inverse amplitude of elliptic integrals, imaginary-argument transformations and two independent periods. His H and theta functions express the resulting elliptic functions as quotients and connect their product and trigonometric-series expansions.\n\nChanging viewpoint from an integral to its inverse makes periodic structure accessible. The complementary modulus connects real and imaginary arguments; H and theta supply numerator and denominator expressions with different shift laws. The same book develops these representations alongside transformations. Legendre and Abel are explicitly acknowledged, and the 1827 announcements and 1829 exposition belong to one contribution family.",
      "title": "Jacobi: elliptic functions, double periods and theta representation"
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