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        "description": "This entry records the dissertation's complex-function and branched-surface framework. Its general mapping argument relies on a variational minimum-attainment step; that step and a modern mapping theorem have not been independently certified here. The inspected source is a corrected German transcription, not the 1851 physical printing.",
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          "citation": "Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (1851); D. R. Wilkins transcription, corrected April 2000",
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      "statement": "# Riemann: complex functions on branched surfaces\n\nRiemann's dissertation organized complex functions through complex differentiability, local angle-preserving maps, and surfaces spread in multiple sheets over the complex plane. Branch points and connectivity became part of the setting for studying functions independently of a formula.\n\nA multivalued expression becomes easier to study when its different values live on connected sheets. Riemann described how circling a branch point moves between sheets and how cuts express the connectivity of the surface. Together with complex differentiability and conformal interpretation, this shifted attention from algebraic expressions to the geometry and boundary behavior of functions. His later geometry lecture and zeta-function paper remain separate contributions.\n\n## Scientific scope\n\nThis entry records the dissertation's complex-function and branched-surface framework. Its general mapping argument relies on a variational minimum-attainment step; that step and a modern mapping theorem have not been independently certified here. The inspected source is a corrected German transcription, not the 1851 physical printing.\n\n## People and systems\n\nBernhard Riemann\n\n\n\n## Original sources\n\n- [Bernhard Riemann, Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (1851); D. R. Wilkins transcription, corrected April 2000](https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/Grund.pdf)\n",
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