In the theory of

ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...

s, a movable singularity is a point where the solution of the equation behaves badly and which is "movable" in the sense that its location depends on the

initial conditions In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...

of the differential equation. Suppose we have an

in the complex domain. Any given solution ''y''(''x'') of this equation may well have singularities at various points (i.e. points at which it is not a regular holomorphic function, such as

branch points In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...

essential singularities In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...

poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in C ...

). A singular point is said to be movable if its location depends on the particular solution we have chosen, rather than being fixed by the equation itself. For example the equation :

\frac = \frac

has solution

y=\sqrt

for any constant ''c''. This solution has a branchpoint at

x=c

, and so the equation has a movable branchpoint (since it depends on the choice of the solution, i.e. the choice of the constant ''c''). It is a basic feature of linear ordinary differential equations that singularities of solutions occur only at singularities of the equation, and so linear equations do not have movable singularities. When attempting to look for 'good' nonlinear differential equations it is this property of linear equations that one would like to see: asking for no movable singularities is often too stringent, instead one often asks for the so-called

Painlevé property Painlevé, a surname, may refer to: __NOTOC__ People * Jean Painlevé (1902–1989), French film director, actor, translator, animator, son Paul * Paul Painlevé (1863–1933), French mathematician and politician, twice Prime Minister of France Mat ...

: 'any movable singularity should be a pole', first used by

Sofia Kovalevskaya Sofya Vasilyevna Kovalevskaya (russian: link=no, Софья Васильевна Ковалевская), born Korvin-Krukovskaya ( – 10 February 1891), was a Russian mathematician who made noteworthy contributions to analysis, partial differen ...

References

* Einar Hille (1997), ''Ordinary Differential Equations in the Complex Domain'', Dover. {{isbn, 0-486-69620-0 Complex analysis Ordinary differential equations