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1. Deferent

   Linked via "Fourier series decomposition"

   Legacy and Conceptual Transition
   The deferent remained the backbone of geocentric modeling until the early 17th century, where it was fundamentally challenged by the work of Kepler. While Kepler's laws demonstrated that planetary paths were elliptical ($e < 1$ for ellipses) rather than circular, the underlying mathematical apparatus developed by Hipparchus and Ptolemy—involving the systematic separation of…

2. Differential Equations

   Linked via "Fourier series"

   Analytical Solutions for PDEs
   The analytical solution of PDEs often requires specialized techniques dependent on the equation's structure. For instance, the Method of Separation of Variables is highly effective for linear, homogeneous PDEs defined on simple geometries (e.g., rectangular domains). This method reduces the PDE into a set of simpler ODEs, whose solutions …