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Of all possible mathematical functions, a few hundred have been given names---like Bessel functions, elliptic integrals, etc. This talk will trace the sometimes surprising 300-or-so-year history of such special functions. It will discuss the impact of Mathematica on the world of such functions, and the significance of the functions for mathematics and mathematical modeling. The talk will also include some new ideas about the boundary of the space of mathematical problems that can be solved exactly, and the significance of constructs such as special functions in defining this boundary.
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