proof_that_0_fact_equals_1

proof_that_0_fact_equals_1

$0! `:=` \Gamma (1)$

$(z) := _{0}^{} t^{z-1}e{-t}dt $

Γ(1) = ∫₀^∞t⁰e^−tdt

 = ∫₀^∞e^−tdt

using the substitution method for integration.

$x `:=` -t$

$\implies \frac{dx}{d t} = -1$

 ⟹ dx = −dt

And the bounds of the integral are now 0 to −∞.

 ⟹ ∫₀^∞e^−tdt = −∫₀^−∞e^xdx

We know that the integral of e^x is itself, e^x.

 = −[e^x]₀^−∞

$ = - (0 - 1) = - (-1) = 1$. QED.

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