Lagmental Vicfred

Beta Is a Quotient of Three Gamma Values by Vicfred

tags: mathematics, special functions, asymptotic analysis, distributions

A change of variables on the positive quadrant relates the beta integral to gamma functions. I will separate the object being defined from the consequence being claimed.

Set-up

Special functions extend \(n!\), integrals, and differential equations beyond integer parameters. Asymptotic notation \(f(x)\sim g(x)\) means their ratio tends to one.

$$ B(x,y)=\int_0^1t^{x-1}(1-t)^{y-1}\,dt $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} $$

The calculation

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ \int_0^\infty\!\int_0^\infty u^{x-1}v^{y-1}e^{-(u+v)}\,du\,dv=\Gamma(x+y)B(x,y) $$

What survives abstraction

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad B(x,y)=\int_0^1t^{x-1}(1-t)^{y-1}\,dt,\\[5pt] \mathsf{C}\;&:\quad B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}. \end{aligned} $$

The boundary

An asymptotic expansion need not converge. Truncating near the smallest term can be useful even when the infinite series diverges.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Mon 27 October 2025. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.