Introduzione alla funzione gamma: tra fattoriale e continuo

a) La funzione gamma, indicata con Γ(z), nasce come estensione naturale del fattoriale ai numeri reali e complessi, permettendo di calcolare “fattoriali di numeri non interi”. Mentre il fattoriale n! è definito solo per interi positivi, Γ(z) estende questa idea grazie alla formula integrale:
Γ(z) = ∫₀⁺∞ t⁽ᶻ⁻¹⁾ e⁻ᵗ dt, definita per Re(z) > 0.
Questa generalizzazione è fondamentale in teoria della probabilità, fisica statistica e analisi matematica.
b) Per ogni numero intero positivo n, si ha la connessione chiave: Γ(n) = (n−1)!, collegando così la gamma al classico fattoriale.
c) Al di là del fattoriale, la gamma appare nella distribuzione esponenziale e nelle sue generalizzazioni,

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