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{{MSC|34M03,34M35|32Sxx}}

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<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
$$\dot z=A(t)z,\quad z\in\CC^n,\ A(t)=\|a_{ij}(t)\|_{i,j=1}^n$$
with coefficients $a_j(\cdot)$, resp., $a_{ij}(\cdot)$ [[Meromorphic 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
$$
 |y_*(t)|\le C|t-t_*|^{-d},\quad\text{resp.,}\quad \|z_*(t)\|\le C|t-t_*|^{-d},\qquad 0<C,d<+\infty
\label{(2)}
$$
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 singular point|irregular singularity]].
===Fuchsian condition===
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.
# 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 $j=1,\dots,n$;
# 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|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]].

<small>
===Notes===

<references/></small>

==Multidimensional generalization==
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]]. 


====Bibliography====

{|
|-
|valign="top"|{{Ref|In}}||valign="top"| E. L. Ince, ''Ordinary Differential Equations'', Dover Publications, New York, 1944, {{MR|0010757}}
|-
|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}} 
|-
|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}}
|-
|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}}
|-
|valign="top"|{{Ref|B}}||valign="top"|  H. Bateman (ed.)  A. Erdélyi (ed.) , ''Higher transcendental functions'' , '''3. Automorphic functions''' , McGraw-Hill  (1955)
|}