Binet’s Function:

Binet's function is associated with the logarithmic gamma function and plays a crucial role in various asymptotic analyses, including Stirling's approximation.

For a positive real number $x$, Binet's function is given by:

$B(x) = \log \Gamma(x) - \left( \left(x - \frac{1}{2}\right)\log x - x + \frac{1}{2}\log(2\pi) \right)$

Here, $\Gamma(x)$ represents the Gamma function, and Binet's function captures the remainder in the approximation of $\log \Gamma(x)$.

Stirling’s Formula:

Stirling’s formula is an asymptotic approximation for the factorial $n!$ or the Gamma function $\Gamma(n+1)$. The standard form is:

$n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \quad \text{as} \quad n \to \infty$

Derivation of Stirling's Formula in the Complex Plane:

To generalize Stirling’s formula to the complex plane, we derive an approximation for the Gamma function $\Gamma(z)$, where $z$ is a complex number.

1. Gamma Function Integral Representation:

The Gamma function can be expressed as:

$\Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt$

This integral representation holds for any complex number $z$ with $\Re(z) > 0$.

2. Approximation via Integral Estimation:

To approximate $\Gamma(z)$, we apply the method of steepest descents (a technique from complex analysis used to approximate integrals). For large $|z|$, the integral for $\Gamma(z)$ is dominated by values of $t$ near $z-1$, leading to the following asymptotic expansion for $\log \Gamma(z)$:

$\log \Gamma(z) \sim \left(z - \frac{1}{2}\right) \log z - z + \frac{1}{2}\log(2\pi) + O\left(\frac{1}{z}\right)$

This gives us a generalization of Stirling’s approximation for $\Gamma(z)$, valid in the complex plane:

$\Gamma(z) \sim \sqrt{2\pi} z^{z-\frac{1}{2}} e^{-z} \quad \text{as} \quad |z| \to \infty, \, |\arg(z)| < \pi$

This approximation works for complex $z$ with the condition that $z$ is not near the negative real axis (to avoid singularities of the Gamma function).

Key Takeaways:

• Binet’s function is the difference between $\log \Gamma(x)$ and the main asymptotic terms of Stirling’s formula.
• Stirling’s formula is generalized to the complex plane using the Gamma function and asymptotic analysis techniques like steepest descents.
• The generalization is valid for large complex $z$, except near the negative real axis due to poles of $\Gamma(z)$.

Do you want more details on this derivation or any clarifications?

5 Related Questions:

1. How is Binet's function used in number theory?
2. What is the role of the Gamma function in complex analysis?
3. Can you explain the method of steepest descents in more detail?
4. How do we handle singularities in the Gamma function's complex plane extension?
5. How does Stirling’s approximation improve for higher-order terms?

Tip:

Stirling's approximation becomes more accurate as $n$ increases, and can be refined by adding more terms from the asymptotic expansion!