Define Binet’s function. Derive the generalization to the complex plane of Stirling’s formula

### 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

Define Binet’s function. Derive the generalization to the complex plane of Stirling’s formula.

Explore the definition of Binet's function, a refinement of Stirling’s approximation, and its application to the Gamma function in the complex plane. This article includes key concepts from complex analysis and asymptotic approximations, with detailed formulas and theorems used in higher mathematics.

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