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Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
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nth-order generalizations of the arithmetic-geometric mean
You may want to check a book by J.M. Borwein and P.B. Borwein titled "Pi and the AGM".
The convergence is quadratic in the sense that the number of settled digits doubles in every turn in the long run …