|
|
| Line 1: |
Line 1: |
| {{no footnotes|date=January 2011}}
| | I am Oscar and I totally dig that title. North Dakota is our birth place. What I love doing is playing baseball but I haven't produced a dime with it. Managing people is what I do and the salary has been really satisfying.<br><br>My weblog ... [http://rivoli.enaiponline.com/user/view.php?id=438251&course=1 std testing at home] |
| [[File:MovingSingularity.png|right|thumb|390px|Solutions to the differential equation <math>\frac{dy}{dx} = \frac{1}{2y}</math> subject to the initial conditions y(0)=0, 1 and 2 (red, green and blue curves respectively). The positions of the moving singularity at x= 0, -1 and -4 is indicated by the vertical lines.]]
| |
| | |
| In the theory of [[ordinary differential equation]]s, a '''movable singularity''' is a point where the solution of the equation [[well-behaved|behaves badly]] and which is "movable" in the sense that its location depends on the [[initial conditions]] of the differential equation.<ref name=BenderOrszag7>
| |
| | |
| {{Cite book | last = Bender | first = Carl M. | authorlink = | coauthors = Orszag, Steven A. | title = Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Series | publisher = Springer | date = 1999 | location = | pages = 7 }}</ref>
| |
| Suppose we have an [[ordinary differential equation]] 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]], [[Essential singularity|essential singularities]] or [[Pole (complex analysis)|poles]]). 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
| |
| | |
| :<math> \frac{dy}{dx} = \frac{1}{2y}</math> | |
| | |
| has solution <math>y=\sqrt{x-c}</math> for any constant ''c''. This solution has a branchpoint at <math>x=c</math>, 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 transcendents|Painlevé property]]: 'any movable singularity should be a pole', first used by [[Sofia Kovalevskaya]].
| |
| | |
| == References == | |
| {{reflist}}
| |
| * Einar Hille (1997), ''Ordinary Differential Equations in the Complex Domain'', Dover. ISBN 0-486-69620-0
| |
| | |
| [[Category:Complex analysis]]
| |
| [[Category:Ordinary differential equations]]
| |
I am Oscar and I totally dig that title. North Dakota is our birth place. What I love doing is playing baseball but I haven't produced a dime with it. Managing people is what I do and the salary has been really satisfying.
My weblog ... std testing at home