A generalized exponential function Ï â¡ ( x ) satisfies the equations
and is strictly increasing when 0 ⤠x ⤠1 . Its inverse Ï â¡ ( x ) is called a generalized logarithm. It, too, is strictly increasing when 0 ⤠x ⤠1 , and
These functions are not unique. The simplest choice is given by
Then
and
Correspondingly,
and
where â is the positive integer determined by the condition
Both Ï â¡ ( x ) and Ï â¡ ( x ) are continuously differentiable.
For further information, see Clenshaw et al. (1986). For Câ generalized logarithms, see Walker (1991). For analytic generalized logarithms, see Kneser (1950).
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