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A notion in the theory of ordinary linear differential equations with an independent complex variable. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808701.png" /> is called a regular singular point of the equation
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A notion in the theory of ordinary linear differential equations with an independent complex variable.  
  
<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/r/r080/r080870/r0808702.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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==Definition of regularity==  
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A point $t_*\in\CC$ is called a regular singular point of the equation $$y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_{n-1}(t)y'+a_n(t)y=0\label{(1)}$$
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or of the system
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$$\dot z=A(t)z,\quad z\in\CC^n,\ A(t)=\|a_{ij}(t)\|_{i,j=1}^n$$
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with coefficients $a_j(\cdot)$, resp., $a_{ij}(t)$ [[Meromorphic function|meromorphic]] at the point $t_*$, if  every solution of the equation (resp., the system) increases no faster than polynomially as $t\to t_*$ in any sector. This means that for any proper sector $\{\alpha<\arg (t-t_*)<\beta\}$ with $\beta-\alpha<\pi$ any solution $y_*(t)$ of the equation (resp., any vector solution $z_*(t)$ of the system) is constrained by an inequality of the form
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$$
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|y_*t)|\le C|t-t_*|^{-d},\quad\text{resp.,}\quad \|z_*(t)\|\le C|t-t_*|^{-d},\qquad 0<C,d<+\infty
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\label{(2)}
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$$
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with suitable constants $C,d$.
  
or of the system
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Regular singularities constitute a simplest type of singularities of multivalued functions, closely analogous to polar singularities of single-valued functions.
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There is a simple condition on the coefficients, called the [[Fuchsian equation|Fuchs condition]], which guarantees that the equation (resp., system) has a regular singularity.
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====Fuchsian condition====
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# The $j$th coefficient $a_j(t)$ of the scalar equation (1) has a pole of order $\leqslant j$ at $t=t_*$: $(t-t_*)^j a_j(t)$ extends holomorphically at the point $t_*$ for all $j=1,\dots,n$;
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# The matrix function $A(t)$ has a pole of order 1 (at worst) at the point $t=t_*$: $(t-t_*)A(t)$ admits extension as a holomorphic matrix function at the point $t_*$.
  
<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/r/r080/r080870/r0808703.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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The key difference between the equation (1) and the system (2) is the necessity of the Fuchsian condition for the regularity: any equation exhibiting a regular singular point satisfies the Fuchsian condition at this point, whereas a system with a pole of order $\geqslant 2$ may well be regular.
  
with analytic coefficients, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808704.png" /> is an isolated singularity of the coefficients and if every solution of (1) or (2) increases no faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808705.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808706.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808707.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808708.png" /> within an arbitrary acute angle with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r0808709.png" />. This last restriction is necessary in view of the fact that in a neighbourhood of a regular singular point the solutions are non-single-valued analytic functions, and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087010.png" /> along an arbitrary curve, they can increase essentially faster than they do when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087011.png" /> over a ray with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087012.png" />.
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==Multidimensional generalization==
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Apart from ordinary linear equations and systems, the notion of a regular singularity exists also in the theory of (integrable) [[Pfaffian system|Pfaffian systems]].  
  
For a singular point of the coefficients of (1) or (2) to be a regular singular point of (1) or (2), it must be a [[Pole (of a function)|pole (of a function)]], and not an [[Essential singular point|essential singular point]], of the coefficients. For equation (1) there is Fuchs' condition: The singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087013.png" /> of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087014.png" /> of equation (1) is a regular singular point of (1) if and only if the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087016.png" />, are holomorphic at zero. In the case of the system (2) there is the following sufficient condition: If the entries of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087017.png" /> have a simple pole at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080870/r08087018.png" />, then this point is a regular singular point of (2).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</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">[2]</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">[3]</TD> <TD valign="top">  A.H.M. Levelt,  "Hypergeometric functions I-IV"  ''Proc. Koninkl. Nederl. Akad. Wet. Ser. A'' , '''64''' :  4  (1961)  pp. 362–403</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Deligne,  "Equations différentielles à points singuliers réguliers" , ''Lect. notes in math.'' , '''163''' , Springer  (1970)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J. Plemelj,  "Problems in the sense of Riemann and Klein" , Wiley  (1964)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.I. Arnol'd,  Yu.S. Il'yashenko,  "Ordinary differential equations" , ''Encycl. Math. Sci.'' , '''1''' , Springer  (Forthcoming)  (Translated from Russian)</TD></TR></table>
 
  
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|valign="top"|{{Ref|Go}}||valign="top"| V.V. Golubev,  "Vorlesungen über Differentialgleichungen im Komplexen",  Deutsch. Verlag Wissenschaft.  (1958)  (Translated from Russian) 
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|valign="top"|{{Ref|In}}||valign="top"| E. L. Ince, "Ordinary Differential Equations", Dover Publications, New York (1944), {{MR|0010757}}
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Revision as of 14:46, 15 April 2012

A notion in the theory of ordinary linear differential equations with an independent complex variable.

Definition of regularity

A point $t_*\in\CC$ is called a regular singular point of the equation $$y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_{n-1}(t)y'+a_n(t)y=0\label{(1)}$$ or of the system $$\dot z=A(t)z,\quad z\in\CC^n,\ A(t)=\|a_{ij}(t)\|_{i,j=1}^n$$ with coefficients $a_j(\cdot)$, resp., $a_{ij}(t)$ meromorphic at the point $t_*$, if every solution of the equation (resp., the system) increases no faster than polynomially as $t\to t_*$ in any sector. This means that for any proper sector $\{\alpha<\arg (t-t_*)<\beta\}$ with $\beta-\alpha<\pi$ any solution $y_*(t)$ of the equation (resp., any vector solution $z_*(t)$ of the system) is constrained by an inequality of the form $$ |y_*t)|\le C|t-t_*|^{-d},\quad\text{resp.,}\quad \|z_*(t)\|\le C|t-t_*|^{-d},\qquad 0<C,d<+\infty \label{(2)} $$ with suitable constants $C,d$.

Regular singularities constitute a simplest type of singularities of multivalued functions, closely analogous to polar singularities of single-valued functions. There is a simple condition on the coefficients, called the Fuchs condition, which guarantees that the equation (resp., system) has a regular singularity.

Fuchsian condition

  1. The $j$th coefficient $a_j(t)$ of the scalar equation (1) has a pole of order $\leqslant j$ at $t=t_*$: $(t-t_*)^j a_j(t)$ extends holomorphically at the point $t_*$ for all $j=1,\dots,n$;
  2. The matrix function $A(t)$ has a pole of order 1 (at worst) at the point $t=t_*$: $(t-t_*)A(t)$ admits extension as a holomorphic matrix function at the point $t_*$.

The key difference between the equation (1) and the system (2) is the necessity of the Fuchsian condition for the regularity: any equation exhibiting a regular singular point satisfies the Fuchsian condition at this point, whereas a system with a pole of order $\geqslant 2$ may well be regular.

Multidimensional generalization

Apart from ordinary linear equations and systems, the notion of a regular singularity exists also in the theory of (integrable) Pfaffian systems.


References

[Go] V.V. Golubev, "Vorlesungen über Differentialgleichungen im Komplexen", Deutsch. Verlag Wissenschaft. (1958) (Translated from Russian)
[In] E. L. Ince, "Ordinary Differential Equations", Dover Publications, New York (1944), MR0010757


Comments

Any second-order equation (1) with three regular singular points can be reduced to the hypergeometric equation. In the case of four regular singular points it can be reduced to Heun's equation [a1], Sect. 15.3, which includes an algebraic form of the Lamé equation. Extensions of the concept to systems of partial differential equations are mentioned in (the editorial comments to) Hypergeometric equation.

References

[a1] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 3. Automorphic functions , McGraw-Hill (1955)
How to Cite This Entry:
Regular singular point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_singular_point&oldid=24352
This article was adapted from an original article by Yu.S. Il'yashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article