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In this Wikipedia article: we have the following about the beta function approximation: Stirling's approximation gives the asymptotic formula:

$$ B(x, y) \sim \sqrt{2\pi} \cdot \frac{x^{x - \frac{1}{2}} y^{y - \frac{1}{2}}}{(x + y)^{x + y - \frac{1}{2}}} \quad \text{as } x \to \infty,\, y \to \infty. $$

If, on the other hand, $x$ is large and $y$ is fixed, then:

$$ B(x, y) \sim \Gamma(y)\,x^{-y}. $$

I have two questions, the first, where can I find a reference that mentions these results? I only know Abramowitz, ; Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables and I didn't see this result there. According to the symbol $\sim$, it means that exists $C_1, C_2>0$ such that

$$ C_1 \sqrt{2\pi} \cdot \frac{x^{x - \frac{1}{2}} y^{y - \frac{1}{2}}}{(x + y)^{x + y - \frac{1}{2}}} \leq B(x, y) \leq C_2 \sqrt{2\pi} \cdot \frac{x^{x - \frac{1}{2}} y^{y - \frac{1}{2}}}{(x + y)^{x + y - \frac{1}{2}}} \quad \text{as } x \to \infty,\, y \to \infty. $$

or is it something else?

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    $\begingroup$ The symbol $\sim$ here means, as usual, that $a\sim b\iff a/b\to1$. These asymptotic formulas immediately follow from the Stirling approximation to the gamma function -- which is probably why you cannot find references to them. $\endgroup$ Commented May 16 at 12:24
  • $\begingroup$ By the definition of limit, $a/b \to 1$ ends up implying $c_1b \leq b \leq c_2 b$, right? Do you know of a good reference that talks about these things? $\endgroup$ Commented May 16 at 12:34
  • $\begingroup$ No, I don't know of such references. $\endgroup$ Commented May 18 at 1:59

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