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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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looking for monotonically increasing functions with range in [0,1]
You could, for example, take the function $f(x) =
\dfrac{1-e^{-x}}{2}$ for $x\ge 0$ and
$f(x) = \dfrac{e^{x}}{2}$ otherwise.
Any other base for the exponent would work as well; 2 might be useful for …