The Liouville equation for the system of N interacting particles can be written: (8.11) ∂ f ( N ) ∂ t + ∑ j = 1 N { ( p j m · ∂ f ( N ) ∂ r j ) + ( X j m · ∂ f ( N ) ∂ p j ) } = − ∑ j = 1 N ( F j · ∂ f ( N ) ∂ p j ) .

Proof of Liouville's Theorem f(z₂) – f(z₁) = 0 ⇒ f(z₁) = f(z₂). Since z₁ and z_2, are arbitrarily chosen, this holds for every points in the complex z-plane.

We can express the integral as an elementary function or prove that it is not elementary. We show that if the problem of integration in finite terms is solvable on a given elementary function field k, then it is solvable in any algebraic extension of k(δ), where δ is a logarithm or exponential of an element of k.