Lagmental Vicfred

Gamma Extends the Factorial by an Integral by Vicfred

tags: mathematics, special functions, asymptotic analysis, distributions

Integration by parts gives the recurrence Gamma of z plus one equals z Gamma of z. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

The data

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

$$ \Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt\qquad(\Re z>0) $$

The first display fixes the mathematical data. I label it \(\mathsf{data}\) mentally, while the next is the \(\mathsf{claim}\); the bridge between them is the displayed \(\Longrightarrow\), not an automatic implication.

$$ \Gamma(z+1)=z\Gamma(z),\qquad\Gamma(n+1)=n! $$

Derivation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \Gamma\!\left(\frac12\right)=\int_0^\infty t^{-1/2}e^{-t}\,dt=2\int_0^\infty e^{-u^2}\,du=\sqrt\pi $$

Invariant content

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad \Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt\qquad(\Re z>0),\\[5pt] \mathsf{C}\;&:\quad \Gamma(z+1)=z\Gamma(z),\qquad\Gamma(n+1)=n!. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \Gamma(z+1)=z\Gamma(z),\qquad\Gamma(n+1)=n! \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Wed 08 June 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.