Extensions of his formula that correct the error were given by Stirling himself and by Jacques Philippe Marie Binet.
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The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s, and was solved at the end of the same decade by Leonhard Euler.
Other procedures, especially function-like procedures will have certain behaviours that in specific invocations may enable some work to be avoided: for instance, the Gamma function, if invoked with an integer parameter, could be converted to a calculation involving integer factorials.
For the case of the Riemann zeros Wu and Sprung and others have shown that the potential can be written implicitly in terms of the Gamma function and zeroth-order Bessel function.
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The first one follows from a change of variable of the integral representation of the Gamma function (Abel, 1823), giving a Mellin transform which can be expressed in different ways as a double integral (Sondow, 2005).