View source for Fuchsian equation

''equation of Fuchsian class''

A linear homogeneous ordinary differential equation in the complex domain,

<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/f/f041/f041880/f0418801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

with analytic coefficients, all singular points of which on the [[Riemann sphere|Riemann sphere]] are regular singular points (cf. [[Regular singular point|Regular singular point]]). Equation (1) belongs to the Fuchsian class if and only if its coefficients have 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/f/f041/f041880/f0418802.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f0418803.png" /> are distinct points and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f0418804.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f0418805.png" />. A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f0418806.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f0418807.png" /> equations belongs to the Fuchsian class if it has 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/f/f041/f041880/f0418808.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f0418809.png" /> are distinct points and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188010.png" /> are constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188011.png" />-dimensional matrices. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188012.png" /> are singular for the equation (1) and the system (2). Fuchs' identity holds for (1):

<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/f/f041/f041880/f04188013.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188014.png" /> are the characteristic exponents at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188016.png" /> those at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188017.png" /> (cf. [[Characteristic exponent|Characteristic exponent]]). Fuchsian equations (and systems) are also called regular equations (systems). This class of equations and systems was introduced by J.L. Fuchs .

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188018.png" /> be the Riemann sphere with punctures at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188019.png" />. Every non-trivial solution of (1) (respectively, every component of a solution of (2)) is an analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188020.png" />. In general, this function is infinite-valued, and all the singular points of (1) (or (2)) are branch points of it of infinite order.

A second-order Fuchsian equation with singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188021.png" /> has 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/f/f041/f041880/f04188022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</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/f/f041/f041880/f04188023.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188024.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188025.png" />. The transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188026.png" /> takes a Fuchsian equation to a Fuchsian equation, with

<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/f/f041/f041880/f04188027.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/f/f041/f041880/f04188028.png" /></td> </tr></table>

and the characteristic exponents at the other singular points are unchanged. By means of such transformations, equation (3) can be reduced to 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/f/f041/f041880/f04188029.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/f/f041/f041880/f04188030.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/f/f041/f041880/f04188031.png" /></td> </tr></table>

A second-order Fuchsian equation with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188032.png" /> singular points is completely determined by specifying the values of the characteristic exponents at these points if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188033.png" />. Using a Möbius transformation the equation can be reduced to the form: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188035.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188037.png" /> (the Euler equation); c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188038.png" /> — the [[Papperitz equation|Papperitz equation]] (or Riemann equation).

A matrix Fuchsian equation has 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/f/f041/f041880/f04188039.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188040.png" /> are distinct points, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188041.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188042.png" />-dimensional matrix function, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188043.png" /> are constant matrices. The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188044.png" /> is called a differential substitution at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188045.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188046.png" /> be a closed curve that starts at a non-singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188047.png" />, is positively oriented and contains only the singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188048.png" /> inside it. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188049.png" /> is a solution of (4) that is holomorphic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188050.png" />, then under analytic continuation along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188051.png" />,

<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/f/f041/f041880/f04188052.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188053.png" /> is a constant matrix, called an integral substitution at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188054.png" />. H. Poincaré (see [[#References|[2]]]) posed the so-called first regular Poincaré problem for a system of the form (4). It consists of the following three problems:

A) to represent the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188055.png" /> in its whole domain of existence;

B) to construct the integral substitutions at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188056.png" />;

C) to give an analytic characterization of the singularities of the solutions.

In particular, solving problem B) enables one to construct the monodromy group of (4). A solution of the Poincaré problem was obtained by I.A. Lappo-Danilevskii [[#References|[3]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188059.png" /> be the hyperlogarithms:

<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/f/f041/f041880/f04188060.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/f/f041/f041880/f04188061.png" /></td> </tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188062.png" /> be the element (germ) at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188063.png" /> of a solution of (4), normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188064.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188065.png" /> be the analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188066.png" /> generated by this element. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188067.png" /> is an entire function of the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188068.png" /> and has a series expansion

<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/f/f041/f041880/f04188069.png" /></td> </tr></table>

which converges uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188070.png" /> on every compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188071.png" />. The integral substitution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188072.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188073.png" /> corresponding to the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188074.png" /> is an entire function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188075.png" /> and has a series expansion

<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/f/f041/f041880/f04188076.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188077.png" /> can be expressed in terms of hyperlogarithms (see [[#References|[3]]], [[#References|[6]]]).

Formulas that give a solution to problem C) have also been obtained (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  J.L. Fuchs,   "Zur Theorie der linearen Differentialgleichungen mit Veränderlichen Koeffizienten"  ''J. Reine Angew. Math.'' , '''66'''  (1866)  pp. 121–160</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  J.L. Fuchs,   "Zur Theorie der linearen Differentialgleichungen mit Veränderlichen Koeffizienten. Ergänzung"  ''J. Reine Angew. Math.'' , '''68'''  (1868)  pp. 354–385</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Poincaré,   "Papers on Fuchsian functions" , Springer  (1985)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.A. Lappo-Danilevskii,   "Applications des fonctions matrices dans la theorie des systèeme des équations différentielles ordinaires lineaires" , Moscow  (1957)  (In Russian; translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.A. Coddington,   N. Levinson,   "Theory of ordinary differential equations" , McGraw-Hill  (1955)  pp. Chapts. 13–17</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.V. Golubev,   "Vorlesungen über Differentialgleichungen im Komplexen" , Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''3''' , Addison-Wesley  (1964)  pp. Part 2  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)</TD></TR></table>



====Comments====
The matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188078.png" /> of equation (5) is also called the local monodromy at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188079.png" /> or the monodromy matrix at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188080.png" /> of the Fuchsian system (4). Riemann posed the problem, the Riemann monodromy problem, of finding for given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188081.png" /> a Fuchsian system with these given monodromy matrices. This problem was essentially solved by J. Plemelj [[#References|[a3]]], G. Birkhoff [[#References|[a4]]], [[#References|[a5]]] and I.A. Lappo-Danilevskii [[#References|[a2]]]. By taking a contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188082.png" /> through all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188084.png" /> and a piecewise-constant matrix function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188085.png" /> (value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188086.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188088.png" />, value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188089.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188091.png" />) the problem can be turned into a [[Riemann–Hilbert problem|Riemann–Hilbert problem]]. The conditions on the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188092.png" /> and the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041880/f04188093.png" /> which are necessary and sufficient for the systems to retain the same monodromy under smooth changes in these parameters take the form of differential equations known as the isomonodromy equations or Schlessinger equations. These equations have links to (completely) integrable systems (cf. [[Integrable system|Integrable system]]) and quantum fields, cf., e.g., [[#References|[a6]]], [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hille,   "Ordinary differential equations in the complex domain" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.A. Lappo-Danilevskii,   "Mémoire sur la théorie des systèmes des équations différentielles linéaires" , Dover, reprint  (1953)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Plemelj,   "Problems in the sense of Riemann and Klein" , Wiley  (1964)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.D. Birkhoff,   "Singular points of ordinary linear differential equations"  ''Trans. Amer. Math. Soc.'' , '''10'''  (1909)  pp. 434–470</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.D. Birkhoff,   "A simplified treatment of the regular singular point"  ''Trans. Amer. Math. Soc.'' , '''11'''  (1910)  pp. 199–202</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.V. Chudnovsky,   "Riemann, monodromy problem, isomonodromy deformations and completely integrable systems"  C. Bardos (ed.)  D. Bessis (ed.) , ''Bifurcation phenomena in mathematical physics and related topics'' , Reidel  (1980)  pp. 385–447</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Jimbo,   T. Miwa,   M. Sato,   "Holonomic quantum fields—the unanticipated link between deformation theory of differential equations and quantum fields"  K. Osterwalder (ed.) , ''Mathematical problems in theoretical physics'' , Springer  (1980)  pp. 119–142</TD></TR></table>