The real Gamma functional equation $\mathrm{\Gamma }(s+1)=s\mathrm{\Gamma }(s)$

Depends on

• The real Gamma function by Euler's integral
• Euler's Gamma integral converges exactly for positive real parameters
• If $u,v$ are differentiable on $[a,b]$ with $u',v'$ integrable, then $\int_a^b u v' = u(b)v(b)-u(a)v(a) - \int_a^b u'v$
• The exponential dominates every fixed nonnegative integer power at $+\infty$
• The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$

Used by

• Gautschi's inequality for the real Gamma function Corollary
• The closed form for the volume of the unit n-ball Corollary
• The elementary recurrences for the real Beta function Corollary
• The unit-ball volume is maximal in dimension five Corollary
• The volume of the unit n-ball tends to zero with dimension Corollary
• Γ(n+1)=n! for every natural number n Corollary
• Replacing the sphere measure by the ball measure in Kirchhoff's formula Counterexample
• Riesz transforms on Euclidean space Definition
• A compact-support Cauchy–Pompeiu calculation Example
• A positive non-log-convex solution of the Gamma functional equation Example
• Flux normalization on every centered sphere Example
• Newtonian potential of radial compact data Example
• Special values of the real Gamma and Beta functions Example
• The two-dimensional logarithmic kernel has unit normalized flux Example
• The unit-ball volumes through dimension eight from the Gamma formula Example
• Sphere integrals of a cylindrical function project to weighted ball integrals Lemma
• The Riesz transform is the principal value of its kernel, with the matching constant Lemma
• Bohr--Mollerup characterisation of the real Gamma function Theorem
• Hölder data give a classical Newtonian solution Theorem
• Kirchhoff's formula in three dimensions Theorem
• Poisson kernel and bounded Dirichlet problem on a half-space Theorem
• Symmetry of the Dirichlet Green function Theorem
• The Bochner–Martinelli formula for C1 functions Theorem
• The forced three-dimensional version as a retarded potential Theorem
• The negative Laplacian of the fundamental solution is the unit Dirac distribution Theorem
• The real Gamma function has one minimum and diverges at both ends of its domain Theorem
• Γ(1/2)=√π by Wallis's product Theorem

Dependency tree · two levels

37 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, §1.2 (standard reference, not scraped)
• K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Lecture 6 §1(iv) (standard reference, not scraped)