Regular singular points (definition)

[Substance D]

Hello,

I am trying to understand the definition of regular point, regular singular point and irregular point

for example, the ode. what would be the r,rs or i points of this?

x^3y'''(x)+3x^2y''(x)+4xy(x)=0

dividing gives the standard form

y''+(3/x)y' + (4/x^2)y=0

So, obviously x can't equal zero, does that make x a regular singular point because x=0 gives rise to a singularity? If so, what does "regular" mean?

Thanks,
David

 

[g_edgar]

it seems you divided wrong, or copied the order of the derivatives wrong, or something.

Yes, 0 is a singular point since at least one coefficient has a pole at 0.

In x^3y'''(x)+3x^2y''(x)+4xy(x)=0 it is a regular singular point ... After dividing we have y'''(x)+(3/x)y''(x)+(4/x^2)y(x)=0 and the order of the pole goes up like this: 0,1,0,2 which is lower than the maximum 0,1,2,3 ...

An example irregular singular point: y'''(x)+(3/x^2)y''(x)+(4/x)y(x)=0 now the pole of order 2 in the y'' term is too large.

The reason for this classification is that at a regular singular point the solutions can be written as series in a nice way. At irregular singular points this usually cannot be done.

 

[Substance D]

Thanks for the reply,

I copied the ODE right, just didn't divide write, good catch though :)

I'm not sure what you mean by "poles".

If something is analytic, it means it can be represented by a series solution, correct?

Thanks,
David

 

[g_edgar]

The idea of "poles" comes from complex analysis. In case of a quotient of polynomials (in lowest terms) the poles are the zeros of the denominator.