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The real Gamma function is smooth and its derivatives are logarithmic moments

Statement

The real Gamma function is smooth on $(0, \infty)$ . For every natural $k$ and every $s > 0$ , $Γ^{(k)} (s) = \int_{0}^{\infty} (\log t)^{k} t^{s - 1} e^{- t} d t$ .

Facts & Assumptions

Given: A compact parameter interval $[a, b] \subset (0, \infty)$ and a natural derivative order $k$ .

[F1]

If an integrand and its parameter derivative are continuous, one slice is absolutely improperly integrable, and on each compact parameter interval the derivative has a nonnegative improperly integrable uniform bound, then the integral is continuously differentiable and its derivative is the integral of the parameter derivative (Differentiation under an improper multiple integral under an integrable derivative bound).

[F2]

For every natural $m$ and real $a > 0$ , $x^{m} / \exp (a x) \to 0$ as $x \to + \infty$ (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ).

[F3]

If a property holds at $0$ and passes from $n$ to $n + 1$ , then it holds for every natural $n$ (The principle of mathematical induction).

[F4]

The Euler integral converges for every positive real parameter (Euler's Gamma integral converges exactly for positive real parameters).

Proof

technique · direct

1.1

On $(0, 1]$ , put $u = - \log t$ . Uniformly for $s \in [a, b]$ , the absolute $k$ th parameter derivative becomes $u^{k} e^{- s u} \leq u^{k} e^{- a u}$ after substitution. By [F2] this is eventually bounded by $e^{- a u / 2}$ and is integrable.

givenF2construct

1.2

On $[1, \infty)$ , $s \leq b$ and $\log t \leq t$ allow $∣ (\log t)^{k} t^{s - 1} e^{- t} ∣$ to be bounded by $t^{m} e^{- t}$ for one natural $m$ depending only on $b, k$ . By [F2] this has an integrable exponential majorant.

givenF2construct

2.1

The case $k = 0$ is the defining integral, convergent by [F4] and absolutely convergent because its integrand is nonnegative. If the displayed formula holds at order $k$ , its integrand and parameter derivative are continuous; steps 1.1 and 1.2, applied at orders $k$ and $k + 1$ , give one absolutely integrable slice and an integrable uniform derivative bound. Thus [F1] differentiates once more. By [F3], the formula holds for every natural $k$ .

step 1.1step 1.2F1F3F4

3.1

Every $s > 0$ lies in a compact interval $[a, b] \subset (0, \infty)$ , so step 2.1 proves the formulas and smoothness throughout the positive axis.

step 2.1∎

Depends on

Used by

The real Gamma function has one minimum and diverges at both ends of its domain Theorem

Dependency tree · two levels

52 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

University of Toronto MAT237Y1, The Gamma Function and the Beta Function, §1.3(a) (standard reference, not scraped)
K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(ii) (standard reference, not scraped)