Binet's Function

Binet's function is used to approximate the logarithm of the Gamma function, $\Gamma(z)$, for complex arguments. It is a refinement of Stirling's approximation, which is primarily applied for large values of $z$. Binet’s formula is given by:

$\log \Gamma(z) = \left(z - \frac{1}{2}\right) \log z - z + \frac{1}{2} \log(2\pi) + \int_0^\infty \frac{2t - \frac{1}{6}}{t^2 + z^2} \, dt$

This function refines the approximation of $\log \Gamma(z)$ by providing a more accurate expansion, particularly useful for complex numbers.

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Stirling’s Formula Generalized to the Complex Plane

Stirling’s formula provides an asymptotic approximation for the factorial of large numbers and is generalized to the Gamma function, $\Gamma(z)$, which extends the factorial function to complex arguments.

Stirling's formula for the Gamma function is:

[ \Gamma(z) \sim \sqrt