What is an irregular singularity and how does it affect Laguerre's equation?

[ognik]

Hi - I didn't really follow the book on Irregular singularities, so I have an example using Laguerre's eqtn - please use this to explain the concept...

In standard form: $ y'' + \frac{1-x}{x}y' + \frac{\lambda}{x}y = 0 $ Clearly there is a normal singularity at x = 0

The book says there is also an irregular singularity at $x = \infty$ ?

I tried putting $z = \frac{1}{x}, \therefore y'' + (z-1)y' + z\lambda = 0$ but as z tends to 0, this becomes $y'' - y' = 0$ which seems OK?

 

[gtoguy97]

Yes, that is correct. An irregular singularity at infinity means that when you substitute $z = \frac{1}{x}$ into the equation, the leading term (in this case, the second order derivative) becomes 0 as the limit of $z$ tends to 0. In your example, $y'' - y' = 0$ is a first order linear ODE which can be solved easily. This indicates an irregular singularity at infinity.