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The branch of the theory of ordinary differential equations in which the solutions are studied from the point of view of the theory of analytic functions. A typical formulation of a problem in the analytic theory of differential equations is this: Given a certain class of differential equations, the solutions of which are all analytic functions of one variable, find the specific properties of the analytic functions that are solutions of this class of equations. In this wide sense, the analytic theory of differential equations includes the theory of algebraic functions, the theory of Abelian integrals, the theory of special functions, etc. Special functions — [[Bessel functions|Bessel functions]], [[Airy functions|Airy functions]], [[Legendre functions|Legendre functions]], [[Laguerre functions|Laguerre functions]], Hermite functions (cf. [[Hermite function|Hermite function]]), Chebyshev functions (cf. [[Chebyshev function|Chebyshev function]]), [[Whittaker functions|Whittaker functions]], Weber functions (cf. [[Weber function|Weber function]]), [[Mathieu functions|Mathieu functions]], hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]), Sonin functions and many other functions — are solutions of linear differential equations with analytic coefficients. | The branch of the theory of ordinary differential equations in which the solutions are studied from the point of view of the theory of analytic functions. A typical formulation of a problem in the analytic theory of differential equations is this: Given a certain class of differential equations, the solutions of which are all analytic functions of one variable, find the specific properties of the analytic functions that are solutions of this class of equations. In this wide sense, the analytic theory of differential equations includes the theory of algebraic functions, the theory of Abelian integrals, the theory of special functions, etc. Special functions — [[Bessel functions|Bessel functions]], [[Airy functions|Airy functions]], [[Legendre functions|Legendre functions]], [[Laguerre functions|Laguerre functions]], Hermite functions (cf. [[Hermite function|Hermite function]]), Chebyshev functions (cf. [[Chebyshev function|Chebyshev function]]), [[Whittaker functions|Whittaker functions]], Weber functions (cf. [[Weber function|Weber function]]), [[Mathieu functions|Mathieu functions]], hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]), Sonin functions and many other functions — are solutions of linear differential equations with analytic coefficients. | ||
==Linear theory.== | ==Linear theory.== | ||
| − | Consider a system of | + | Consider a system of $n$ equations in matrix notation: |
| − | + | $$ | |
| + | \dot{x} = A(t) x + f(t) . | ||
| + | \tag{1} | ||
| + | $$ | ||
| − | 1) Let the matrices | + | 1) Let the matrices $A(t)$, $f(t)$ be holomorphic in a region $G \subset \mathbf{C}(t)$, where $\mathbf{C}(t)$ is the complex $t$-plane. Any solution of the system \eqref{1} will then be analytic in $G$ (but will not, in general, be single-valued if $G$ is not simply-connected). It is assumed that $A(t)$ is meromorphic in $G$, and one considers the homogeneous system |
| − | + | $$ | |
| + | \dot{x} = A(t) x . | ||
| + | \tag{2} | ||
| + | $$ | ||
| − | (The matrix | + | (The matrix $A(t)$ is called holomorphic (meromorphic) in $G$ if all its elements are holomorphic (meromorphic) in $G$.) A point $t_0\in G$ is called a pole of the matrix $A(t)$ of order $\nu\ge 1$ if, in a given neighbourhood of this point, |
| − | + | $$ | |
| + | A(t) = A_{-\nu} (t-t_0)^{-\nu} + \cdots + A_{-1} (t-t_0)^{-1} + B(t) , | ||
| + | $$ | ||
| − | where | + | where $A_{-j}$ are constant matrices, $A_{-\nu}\ne 0$, and the matrix $B(t)$ is holomorphic at $t_0$. A pole $t_0\ne \infty$ of order $\nu$ is called a regular singular point if $\nu=1$ and an irregular singular point if $\nu\ge 1$. The case $t_0 = \infty$ is reduced to the case $t_0=0$ by the change of variables $t\mapsto t^{-1}$. In what follows, $t_0 \ne 0$. |
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246031.png" /> be a pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246032.png" />. Then there exists a fundamental matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246033.png" /> for the system (2) of the form | 2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246031.png" /> be a pole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246032.png" />. Then there exists a fundamental matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246033.png" /> for the system (2) of the form | ||
Revision as of 18:24, 1 August 2015
The branch of the theory of ordinary differential equations in which the solutions are studied from the point of view of the theory of analytic functions. A typical formulation of a problem in the analytic theory of differential equations is this: Given a certain class of differential equations, the solutions of which are all analytic functions of one variable, find the specific properties of the analytic functions that are solutions of this class of equations. In this wide sense, the analytic theory of differential equations includes the theory of algebraic functions, the theory of Abelian integrals, the theory of special functions, etc. Special functions — Bessel functions, Airy functions, Legendre functions, Laguerre functions, Hermite functions (cf. Hermite function), Chebyshev functions (cf. Chebyshev function), Whittaker functions, Weber functions (cf. Weber function), Mathieu functions, hypergeometric functions (cf. Hypergeometric function), Sonin functions and many other functions — are solutions of linear differential equations with analytic coefficients.
Linear theory.
Consider a system of $n$ equations in matrix notation:
$$ \dot{x} = A(t) x + f(t) . \tag{1} $$
1) Let the matrices $A(t)$, $f(t)$ be holomorphic in a region $G \subset \mathbf{C}(t)$, where $\mathbf{C}(t)$ is the complex $t$-plane. Any solution of the system \eqref{1} will then be analytic in $G$ (but will not, in general, be single-valued if $G$ is not simply-connected). It is assumed that $A(t)$ is meromorphic in $G$, and one considers the homogeneous system
$$ \dot{x} = A(t) x . \tag{2} $$
(The matrix $A(t)$ is called holomorphic (meromorphic) in $G$ if all its elements are holomorphic (meromorphic) in $G$.) A point $t_0\in G$ is called a pole of the matrix $A(t)$ of order $\nu\ge 1$ if, in a given neighbourhood of this point,
$$ A(t) = A_{-\nu} (t-t_0)^{-\nu} + \cdots + A_{-1} (t-t_0)^{-1} + B(t) , $$
where $A_{-j}$ are constant matrices, $A_{-\nu}\ne 0$, and the matrix $B(t)$ is holomorphic at $t_0$. A pole $t_0\ne \infty$ of order $\nu$ is called a regular singular point if $\nu=1$ and an irregular singular point if $\nu\ge 1$. The case $t_0 = \infty$ is reduced to the case $t_0=0$ by the change of variables $t\mapsto t^{-1}$. In what follows, $t_0 \ne 0$.
2) Let 
be a pole of 
. Then there exists a fundamental matrix 
for the system (2) of the form
 | (3) |
where 
is a constant matrix, 
is holomorphic if 
where 
is a regular singular point, and 
is holomorphic if 
where 
is an irregular singular point, for some 
. (Here, 
, by definition.) For a regular singular point the matrix 
can be expressed in terms of 
in an explicit form [1], [2]; this is not the case for irregular singular points.
A similar classification of singular points is introduced for differential equations of order 
with meromorphic coefficients. Differential equations and differential systems all singular points of which are regular are known as Fuchsian differential equations (systems). The general form of 
for such a system is:
 |
An example of a Fuchsian differential equation is the hypergeometric equation.
3) Let 
where 
is an integer and let 
be holomorphic at 
(
is an irregular singular point if 
). If 
is a sufficiently narrow sector of the form 
, 
, then there exists a fundamental matrix of the form
 | (4) |
where 
is a constant matrix, 
is a diagonal matrix whose elements are polynomials in 
, 
is an integer and
 |
as 
. The plane 
is subdivided into a finite number of sectors, and in of them there exists a fundamental matrix of the form (4) ([3], [4]; see also [1], [2]).
4) As a result of analytic continuation along a closed path 
the fundamental matrix 
is multiplied by 
, where 
is a constant matrix; one obtains the monodromy group of the differential equation. I.A. Lappo-Danilevskii [5] has studied the problem of Riemann: Let 
be a rational function of 
and let the singularities of the fundamental matrix 
be known, find 
.
5) Let the function 
be a conformal mapping of the upper half-plane 
onto the interior of a polygon, the boundary of which consists of a finite number of segments of straight lines and circular arcs. The function 
will then satisfy the Schwarz equation:
 | (5) |
where 
is a rational function, and the equation
 | (6) |
is Fuchsian. Any solution of equation (5) may be represented in the form 
, where 
and 
are linearly independent solutions of equation (6). Let 
be an infinite discrete group and let 
be an automorphic function of 
, then 
can be represented as 
, where 
are linearly independent solutions of equation (6) and 
is some algebraic function.
Non-linear theory.
1) Consider the Cauchy problem:
 | (7) |
where 
, 
.
Cauchy's theorem: Let the function 
be holomorphic in 
in a region 
and let the point 
. Then there exists a 
such that in the domain 
there exists a solution 
of the Cauchy problem (7), which is unique and holomorphic.
An analytic continuation of the solution 
will also be a solution of the system (7), but the function obtained as a result of the continuation may have singularities and, in the general case, is a many-valued function of 
. The problems which arise are: What singularities may this function have and how can one construct the general solution? In the linear case these questions have been conclusively answered. In the non-linear case the situation is much more complicated and has not been fully clarified even when the 
are rational functions of 
.
2) Consider the differential equation:
 | (8) |
where 
and 
and 
are holomorphic functions of 
in a certain region 
. A point 
is called an (essentially) singular point of equation (8) if 
, 
. Below the structure of the solutions in a neighbourhood of a singular point of the equation is clarified. Develop 
and 
into Taylor series:
 |
 |
and let 
be the eigen values of the matrix 
. The following theorem holds. Let 
and let none of the numbers 
be either a non-negative integer or a real negative number. Then there exists a neighbourhood 
of the point 
, a neighbourhood 
of the point 
, and functions 
and 
such that the mapping 
defined by these functions is biholomorphic, and the differential equation (8) in the new variables assumes the form [6]:
 |
All solutions of equation (8) in the new variables are written in the form 
and 
. Thus, a singular point of the equation is a branching point of infinite order for all solutions of equation (8) (except for the trivial solutions). The singular points of the solution which coincide with the singular points of the equation are called stationary. As distinct from the linear case, the solution of a non-linear equation may have singular points not only at the singular points of the equation; such singular points of the solution are called movable. Painlevé's theorem is valid: The solutions of the equation
 |
where 
is a polynomial in 
and 
with holomorphic coefficients in 
, has no movable transcendental singular points [7].
If, in equation (8), 
and 
are polynomials in 
, then, in view of Painlevé's theorem, all movable singular points are algebraic. On substituting 
, 
, equation (8) assumes the form
 |
where 
and 
are polynomials. Let 
be the roots of the equation 
. The points 
are called infinitely-remote singular points of equation (8); the structure of the solutions in a neighbourhood of these points is described by the theorem quoted above [6].
Let 
and 
be polynomials of degree 
. Since 
are defined by their coefficients and the pair 
defines the same equation, one obtains a one-to-one correspondence between equations (8) and the points of the complex projective space 
, 
. The following theorem is valid: If some set of measure zero is removed from 
, the remaining equations (8) will have the following property: All solutions 
are everywhere dense in 
[8].
3) Consider the autonomous system
 | (9) |
. A point 
will then be a singular point of the system (9) if 
. Poincaré's theorem is valid: Let 
be a singular point of the autonomous system (9). Also let a) the elementary divisors of the Jacobi matrix 
be prime divisors; and b) the eigen values 
of this matrix lie on one side of some straight line in 
passing through the coordinate origin. Then there exists neighbourhoods 
of the points 
and a biholomorphic mapping 
such that the system (9) expressed in the variable 
assumes the form [9]:
 |
If only condition a) is satisfied, it is possible, by using a transformation 
, where 
is a formal power series, to convert system (9) in a neighbourhood of a singular point into a system which can be integrated in quadratures [9], [10]. However, the convergence of these series has been proved on assumptions close to a) and b). If the function 
and the transformation 
are real for real 
, a theorem similar to the theorem of Poincaré has been proved [11]. The structure of the solutions of the autonomous system (9) in general, where 
are polynomials and 
, has not yet (1970s) been studied.
References
| [1] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) MR0069338 Zbl 0064.33002 |
| [2] | W. Wazov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965) |
| [3] | G.D. Birkhoff, "Singular points of ordinary linear differential equations" Trans. Amer. Math. Soc. , 10 (1909) pp. 436–470 MR1500848 Zbl 46.0695.02 Zbl 45.0484.01 Zbl 44.0373.01 |
| [4] | W.J. Trjitzinsky, "Analytic theory of linear differential equations" Acta Math. , 62 (1934) pp. 167–226 MR1555383 Zbl 60.1109.01 |
| [5] | J.A. Lappo-Danilevsky, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint (1953) |
| [6] | L. Bieberbach, "Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt" , Springer (1965) MR0176133 Zbl 0124.04603 |
| [7] | V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) MR0100119 |
| [8] | M.G. Khudai-Verenov, "On a property of solutions of a differential equation" Mat. Sb. , 56 (98) : 3 (1962) pp. 301–308 (In Russian) |
| [9] | V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502 |
| [10] | A.D. Bryuno, "Local methods in nonlinear differential equations" , Springer (1989) (Translated from Russian) MR0993771 |
| [11] | C.L. Siegel, "Ueber die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung" Nachrichten Akad. Wissenschaft. Göttingen (1952) pp. 21–30 Zbl 0047.32901 |
| [12] | H. Poincaré, , Oeuvres de H. Poincaré , 3 , Gauthier-Villars (1916–1965) |
| [13] | L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951) MR1522111 Zbl 55.0810.04 Zbl 46.0621.01 Zbl 45.0693.07 |
Comments
The Riemann monodromy problem mentioned above is of great importance in the modern theory of completely integrable or soliton equations. Cf. Soliton.
References
| [a1] | E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) MR1570308 MR0010757 MR1524980 MR0000325 MR1522581 Zbl 0612.34002 Zbl 0191.09801 Zbl 0063.02971 Zbl 0022.13601 Zbl 65.1253.02 Zbl 53.0399.07 |
| [a2] | E. Hille, "Ordinary differential equations in the complex domain" , Wiley (Interscience) (1976) |
Analytic theory of differential equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_theory_of_differential_equations&oldid=24367