Eigenfunctions of the Laplacian are special functions that arise in the study of differential equations on manifolds, characterized by their relationship with the Laplace operator. When applied to a function, the Laplacian produces a scalar multiple of that function, known as the eigenvalue. These eigenfunctions play a crucial role in various applications, such as understanding the geometry and topology of manifolds, as well as analyzing heat diffusion and vibrations in physical systems.
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