View source for Analytic theory of differential equations

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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.

==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 $t_0$ be a pole of $A(t)$. Then there exists a fundamental matrix $X(t)$ for the system \eqref{2} of the form

$$
  X(t) = \Phi(t) (t-t_0)^D
\tag{3}
$$

where $D$ is a constant matrix, $\Phi(t)$ is holomorphic for $|t-t_0|<\rho$ when $t_0$ is a regular singular point, and $\Phi(t)$ is holomorphic for $0<|t-t_0|<\rho$ when $t_0$ is an irregular singular point, for some $\rho > 0$. (Here, $(t-t_0)^D = \exp(D\ln(t-t_0))$, by definition.) For a regular singular point the matrix $D$ can be expressed in terms of $A(t)$ in an explicit form [[#References|[1]]], [[#References|[2]]]; this is not the case for irregular singular points.

A similar classification of singular points is introduced for differential equations of order $n$ with meromorphic coefficients. Differential equations and systems of differential equations with only regular singular points are known as Fuchsian (systems of) differential equations. The general form of $A(t)$ for such a system is:

$$
  A(t) = \sum_{j=1}^{k} (t-t_j)^{-1} A_j,
  \quad A_j = \text{const},
  \quad k < \infty .
$$

An example of a Fuchsian differential equation is the [[Hypergeometric equation|hypergeometric equation]].

3) Let $A(t) = t^q B(t)$ where $q\ge 0$ is an integer and $B(t)$ is holomorphic at $t=\infty$ ($\infty$ is an irregular singular point if $B(\infty) \ne 0$). If $S$ is a sufficiently narrow sector of the form $|t| > R$, $\alpha < \arg t < \beta$, then there exists a fundamental matrix of the form

$$
  X(t) = P(t) t^Q \exp(R(t)) ,
\tag{4}
$$

where $Q$ is a constant matrix, $R(t)$ is a diagonal matrix whose elements are polynomials in $t^{1/p}$, $p\ge 1$ is an integer and

$$
  P(t) \sim \sum_{j=0}^\infty P_j t^{-j/p}
$$

as $|t| \to \infty$, $t\in S$. The plane $\mathbf{C}(t)$ is subdivided into a finite number of sectors, and in each of them there exists a fundamental matrix of the form \eqref{4} ([[#References|[3]]], [[#References|[4]]]; see also [[#References|[1]]], [[#References|[2]]]).

4) As a result of analytic continuation along a closed path $\gamma$ the fundamental matrix $X(t)$ is multiplied by a constant matrix $B_\gamma$: $X(t) \mapsto X(t) B_\gamma$; one obtains the [[Monodromy group|monodromy group]] of the differential equation. I.A. Lappo-Danilevskii [[#References|[5]]] has studied the problem of Riemann: Let $A(t)$ be a rational function of $t$ and let the singularities of the fundamental matrix $X(t)$ be known, find $A(t)$.

5) Let the function $z=\phi(t)$ be a conformal mapping of the upper half-plane $\operatorname{Im} t > 0$ 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 $\phi(t)$ will then satisfy the Schwarz equation:

$$
  \{ z , t \} \equiv \frac{z'''}{z'} - \frac{3}{2} \left( \frac{z''}{z'} \right)^2 = R(t) .
\tag{5}
$$

where $R(t)$ is a rational function, and the equation

$$
  w'' + \frac{1}{2} R(t) w = 0
\tag{6}
$$

is Fuchsian. Any solution of equation \eqref{5} may be represented in the form $z = w_1 / w_2$, where $w_1$ and $w_2$ are linearly independent solutions of equation \eqref{6}. Let $G$ be an infinite discrete group and let $\phi(t)$ be an [[Automorphic function|automorphic function]] of $G$, then $\phi(t)$ can be represented as $\phi = w_1 / w_2$, where $w_1$, $w_2$ are linearly independent solutions of equation \eqref{6} and $R(t)$ is some [[Algebraic function|algebraic function]].

==Non-linear theory.==


1) Consider the Cauchy problem:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246090.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246092.png" />.

Cauchy's theorem: Let the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246093.png" /> be holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246094.png" /> in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246095.png" /> and let the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246096.png" />. Then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246097.png" /> such that in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246098.png" /> there exists a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a01246099.png" /> of the Cauchy problem (7), which is unique and holomorphic.

An analytic continuation of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460100.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460101.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460102.png" /> are rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460103.png" />.

2) Consider the differential equation:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460107.png" /> are holomorphic functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460108.png" /> in a certain region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460109.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460110.png" /> is called an (essentially) singular point of equation (8) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460112.png" />. Below the structure of the solutions in a neighbourhood of a singular point of the equation is clarified. Develop <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460114.png" /> into Taylor series:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460115.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460116.png" /></td> </tr></table>

and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460117.png" /> be the eigen values of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460118.png" />. The following theorem holds. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460119.png" /> and let none of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460120.png" /> be either a non-negative integer or a real negative number. Then there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460121.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460122.png" />, a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460123.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460124.png" />, and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460125.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460126.png" /> such that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460127.png" /> defined by these functions is biholomorphic, and the differential equation (8) in the new variables assumes the form [[#References|[6]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460128.png" /></td> </tr></table>

All solutions of equation (8) in the new variables are written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460129.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460130.png" />. 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460131.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460132.png" /> is a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460133.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460134.png" /> with holomorphic coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460135.png" />, has no movable transcendental singular points [[#References|[7]]].

If, in equation (8), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460136.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460137.png" /> are polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460138.png" />, then, in view of Painlevé's theorem, all movable singular points are algebraic. On substituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460139.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460140.png" />, equation (8) assumes the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460141.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460143.png" /> are polynomials. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460144.png" /> be the roots of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460145.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460146.png" /> 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 [[#References|[6]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460148.png" /> be polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460149.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460150.png" /> are defined by their coefficients and the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460151.png" /> defines the same equation, one obtains a one-to-one correspondence between equations (8) and the points of the complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460153.png" />. The following theorem is valid: If some set of measure zero is removed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460154.png" />, the remaining equations (8) will have the following property: All solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460155.png" /> are everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460156.png" /> [[#References|[8]]].

3) Consider the autonomous system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460157.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460158.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460159.png" /> will then be a singular point of the system (9) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460160.png" />. Poincaré's theorem is valid: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460161.png" /> be a singular point of the autonomous system (9). Also let a) the elementary divisors of the Jacobi matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460162.png" /> be prime divisors; and b) the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460163.png" /> of this matrix lie on one side of some straight line in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460164.png" /> passing through the coordinate origin. Then there exists neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460165.png" /> of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460166.png" /> and a biholomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460167.png" /> such that the system (9) expressed in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460168.png" /> assumes the form [[#References|[9]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460169.png" /></td> </tr></table>

If only condition a) is satisfied, it is possible, by using a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460170.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460171.png" /> 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 [[#References|[9]]], [[#References|[10]]]. However, the convergence of these series has been proved on assumptions close to a) and b). If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460172.png" /> and the transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460173.png" /> are real for real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460174.png" />, a theorem similar to the theorem of Poincaré has been proved [[#References|[11]]]. The structure of the solutions of the autonomous system (9) in general, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460175.png" /> are polynomials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012460/a012460176.png" />, has not yet (1970s) been studied.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) {{MR|0069338}} {{ZBL|0064.33002}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> W. Wazov, "Asymptotic expansions for ordinary differential equations" , Interscience (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> G.D. Birkhoff, "Singular points of ordinary linear differential equations" ''Trans. Amer. Math. Soc.'' , '''10''' (1909) pp. 436–470 {{MR|1500848}} {{ZBL|46.0695.02}} {{ZBL|45.0484.01}} {{ZBL|44.0373.01}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W.J. Trjitzinsky, "Analytic theory of linear differential equations" ''Acta Math.'' , '''62''' (1934) pp. 167–226 {{MR|1555383}} {{ZBL|60.1109.01}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> J.A. Lappo-Danilevsky, "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Chelsea, reprint (1953)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> L. Bieberbach, "Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt" , Springer (1965) {{MR|0176133}} {{ZBL|0124.04603}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian) {{MR|0100119}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> M.G. Khudai-Verenov, "On a property of solutions of a differential equation" ''Mat. Sb.'' , '''56 (98)''' : 3 (1962) pp. 301–308 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) {{MR|0121520}} {{ZBL|0089.29502}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> A.D. Bryuno, "Local methods in nonlinear differential equations" , Springer (1989) (Translated from Russian) {{MR|0993771}} {{ZBL|}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> C.L. Siegel, "Ueber die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung" ''Nachrichten Akad. Wissenschaft. Göttingen'' (1952) pp. 21–30 {{MR|}} {{ZBL|0047.32901}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> H. Poincaré, , ''Oeuvres de H. Poincaré'' , '''3''' , Gauthier-Villars (1916–1965)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> L.R. Ford, "Automorphic functions" , Chelsea, reprint (1951) {{MR|1522111}} {{ZBL|55.0810.04}} {{ZBL|46.0621.01}} {{ZBL|45.0693.07}} </TD></TR></table>



====Comments====
The Riemann monodromy problem mentioned above is of great importance in the modern theory of completely integrable or soliton equations. Cf. [[Soliton|Soliton]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) {{MR|1570308}} {{MR|0010757}} {{MR|1524980}} {{MR|0000325}} {{MR|1522581}} {{ZBL|0612.34002}} {{ZBL|0191.09801}} {{ZBL|0063.02971}} {{ZBL|0022.13601}} {{ZBL|65.1253.02}} {{ZBL|53.0399.07}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E. Hille, "Ordinary differential equations in the complex domain" , Wiley (Interscience) (1976)</TD></TR></table>