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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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{{TEX|done}}
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{{MSC|34M03,34M35|32Sxx}}
  
<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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A notion in the theory of ordinary linear differential equations with an independent complex variable. A point in the plane of the independent variable is ''regular singular'', if solutions of the equation loose analyticity, but exhibit at most polynomial growth at this point.  
  
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==Regularity as a growth condition for solutions==
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A point $t_*\in\CC$ is called a ''regular'' singular<ref>The construct "regular singularity", which is an [https://en.wikipedia.org/wiki/Oxymoron oxymoron], is too firmly rooted to be replaced by terms like "moderate" or "tame" singularity as was suggested in {{Cite|IY}}. The regular singular point should not be confused with a regular (nonsingular) point at which the coefficients $a_j(t)$, resp., $a_{jk}(t)$, are holomorphic.</ref>  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
 
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., $a_{ij}(\cdot)$ [[Meromorphic function|meromorphic]]<ref>In particular, the coefficients should be holomorphic in a punctured neighborhood of $t_*$ and at worst a finite order pole at it.</ref> 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 $C,d$. The point $t_*=\infty$ is regular, if the equation (resp., the system) has a regular singularity at the point $\tau=0$ after the change of the independent variable $t=1/\tau$.
  
<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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Regular singularities constitute a simplest type of singularities of multivalued functions, closely analogous to polar singularities of single-valued functions.  
  
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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A singular point (a pole of coefficients) which is not regular, usually referred to as an [[Irregular singular point|irregular singularity]].
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===Fuchsian condition===
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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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# 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 $t=t_*$: the product $(t-t_*)A(t)$ admits extension as a holomorphic matrix function at the point $t_*$.
  
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).
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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.
  
====References====
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====Example: Euler equation, Euler system====
<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  (1961pp. 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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The equation $t^n y^{(n)}+c_1t^{n-1}\,y^{(n-1)}+\cdots+c_{n-1}\,ty'+c_n\, y=0$ with constant coefficients $c_1,\dots,c_n\in\CC$ is Fuchsian at the points $t=0,\infty$ (and nonsingular at all other points). The system $t\dot z=A z$ with a constant $n\times n$-matrix $A$ has two Fuchsian singular points at $t=0,\infty$.
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===Special cases===
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Any second-order $(n=2)$ equation (1) with three regular singular points on the Riemann sphere $\CC\cup\infty$ can be reduced to the [[Hypergeometric equation|hypergeometric equation]]. In the case of four regular singular points it can be reduced to Heun's  equation {{Cite|B|Sect. 15.3}}, which includes an algebraic form of the [[Lamé equation|Lamé equation]]. Extensions of the concept to systems of partial differential equations are mentioned in (the  editorial comments to) [[Hypergeometric equation|Hypergeometric  equation]].
  
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<small>
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===Notes===
  
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<references/></small>
  
====Comments====
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==Multidimensional generalization==
Any second-order equation (1) with three regular singular points can be reduced to the [[Hypergeometric equation|hypergeometric equation]]. In the case of four regular singular points it can be reduced to Heun's equation [[#References|[a1]]], Sect. 15.3, which includes an algebraic form of the [[Lamé equation|Lamé equation]]. Extensions of the concept to systems of partial differential equations are mentioned in (the editorial comments to) [[Hypergeometric equation|Hypergeometric equation]].
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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]], see [[local system|local systems]].  
  
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)</TD></TR></table>
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====Bibliography====
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{|
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|-
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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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|-
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|valign="top"|{{Ref|H}}||valign="top"| P. Hartman,''Ordinary differential equations'', Classics in Applied Mathematics '''38''', Corrected reprint of the second (1982) edition,  SIAM Publ., Philadelphia, PA, 2002, {{MR|0658490}},  {{MR|1929104}}
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|-
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|valign="top"|{{Ref|D}}||valign="top"| P. Deligne, ''Équations différentielles à points singuliers réguliers'', (French) Lecture Notes in Mathematics, Vol. '''163'''. Springer-Verlag, Berlin-New York,  1970 {{MR|0417174}}
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|-
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|valign="top"|{{Ref|IY}}||valign="top"| Yu. Ilyashenko, S. Yakovenko, ''Lectures on analytic differential equations''. Graduate Studies in Mathematics, '''86'''. American Mathematical Society, Providence, RI,  2008 {{MR|2363178}}
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|-
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|valign="top"|{{Ref|B}}||valign="top"| H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)
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|}

Latest revision as of 19:19, 27 July 2024

2020 Mathematics Subject Classification: Primary: 34M03,34M35 Secondary: 32Sxx [MSN][ZBL]

A notion in the theory of ordinary linear differential equations with an independent complex variable. A point in the plane of the independent variable is regular singular, if solutions of the equation loose analyticity, but exhibit at most polynomial growth at this point.

Regularity as a growth condition for solutions

A point $t_*\in\CC$ is called a regular singular[1] 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., $a_{ij}(\cdot)$ meromorphic[2] 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 $C,d$. The point $t_*=\infty$ is regular, if the equation (resp., the system) has a regular singularity at the point $\tau=0$ after the change of the independent variable $t=1/\tau$.

Regular singularities constitute a simplest type of singularities of multivalued functions, closely analogous to polar singularities of single-valued functions.

A singular point (a pole of coefficients) which is not regular, usually referred to as an irregular singularity.

Fuchsian condition

There is a simple condition on the coefficients, called the Fuchs condition, which guarantees that the equation (resp., system) has a regular singularity.

  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 $t=t_*$: the product $(t-t_*)A(t)$ 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.

Example: Euler equation, Euler system

The equation $t^n y^{(n)}+c_1t^{n-1}\,y^{(n-1)}+\cdots+c_{n-1}\,ty'+c_n\, y=0$ with constant coefficients $c_1,\dots,c_n\in\CC$ is Fuchsian at the points $t=0,\infty$ (and nonsingular at all other points). The system $t\dot z=A z$ with a constant $n\times n$-matrix $A$ has two Fuchsian singular points at $t=0,\infty$.

Special cases

Any second-order $(n=2)$ equation (1) with three regular singular points on the Riemann sphere $\CC\cup\infty$ can be reduced to the hypergeometric equation. In the case of four regular singular points it can be reduced to Heun's equation [B, 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.

Notes

  1. The construct "regular singularity", which is an oxymoron, is too firmly rooted to be replaced by terms like "moderate" or "tame" singularity as was suggested in [IY]. The regular singular point should not be confused with a regular (nonsingular) point at which the coefficients $a_j(t)$, resp., $a_{jk}(t)$, are holomorphic.
  2. In particular, the coefficients should be holomorphic in a punctured neighborhood of $t_*$ and at worst a finite order pole at it.

Multidimensional generalization

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


Bibliography

[In] E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, MR0010757
[H] P. Hartman,Ordinary differential equations, Classics in Applied Mathematics 38, Corrected reprint of the second (1982) edition, SIAM Publ., Philadelphia, PA, 2002, MR0658490, MR1929104
[D] P. Deligne, Équations différentielles à points singuliers réguliers, (French) Lecture Notes in Mathematics, Vol. 163. Springer-Verlag, Berlin-New York, 1970 MR0417174
[IY] Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178
[B] 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=11770
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