Hermite theorem

Hermite's theorem is a statement about the properties of solutions of first order differential equations in which the independent variable does not belong.

Content

Formulation

If the equation of the first order, which does not include the independent variable $z$ ${\ displaystyle z}$ (i.e. $P(\omega ^{'},\omega )=0$ ${\ displaystyle P (\ omega ^ {'}, \ omega) = 0}$ algebraic with respect to the unknown function and its derivatives, i.e. $P$ ${\ displaystyle P}$ - relatively polynomial $\omega$ ${\ displaystyle \ omega}$ and $\omega ^{'}$ ${\ displaystyle \ omega ^ {'}}$ ) has no critical moving points, its genre is equal to or $0$ ${\ displaystyle 0}$ or $one$ ${\ displaystyle 1}$ . In this case, the integral of the equation is either a rational function , or rationally expressed through exponential or elliptic functions.

Explanation

A special point is the point where the analyticity of the function of a complex variable is violated ^[1] . If the function changes its value when traversing around a singular point, then the singular point is called the critical point ^[2] . A singular point of an integral whose position does not depend on the initial data defining the integral is called a fixed singular point and a singular point whose position depends on the initial data defining the integral is called a movable singular point ^[3] .

Proof

The proof of the Hermite theorem takes $2$ ${\ displaystyle 2}$ pages in the book ^[4] .

Notes

↑ Methods of the theory of functions of a complex variable, 1958 , p. 91.
Lectures on the analytic theory of differential equations, 1941 , p. 35
Lectures on the analytic theory of differential equations, 1941 , p. 41
Lectures on the analytic theory of differential equations, 1941 , p. 100-101.

Literature

Golubev V. V. Lectures on the analytic theory of differential equations. - M.-L .: GOSTEKHTEORIZDAT, 1941. - 400 p.
Lavrentyev MA , Shabat B.V. Methods of the theory of functions of a complex variable. - M .: Fizmatlit, 1958. - 678 p.

[_88b2553e2d7710e0-1] Methods of the theory of functions of a complex variable, 1958 , p. 91.

[_6ada259a3ab68fa7-2] Lectures on the analytic theory of differential equations, 1941 , p. 35

[_6ada209a3ab68722-3] Lectures on the analytic theory of differential equations, 1941 , p. 41

[_862133cfc3163313-4] Lectures on the analytic theory of differential equations, 1941 , p. 100-101.