Question

State and prove the
Liouville's theorem in its standard from.​

Answer 1

In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841,[1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions.

In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841,[1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions.The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is {\displaystyle e^{-x^{2}},}e^{{-x^{2}}}, whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions {\displaystyle {\frac {\sin(x)}{x}}}{\frac {\sin(x)}{x}} and {\displaystyle x^{x}}x^{x}.

In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841,[1][2][3] places an important restriction on antiderivatives that can be expressed as elementary functions.The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is {\displaystyle e^{-x^{2}},}e^{{-x^{2}}}, whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions {\displaystyle {\frac {\sin(x)}{x}}}{\frac {\sin(x)}{x}} and {\displaystyle x^{x}}x^{x}.Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.

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