Why derive Regular Singular Points?

In summary, the Bessel and Legendre equations, which are obtained from writing the Laplacian in cylindrical and spherical coordinates respectively, are important in physics because the Laplacian is a common second order differential operator that is invariant under rigid motions and shows up in many physical problems such as potential energy, wave equations, and heat or diffusion equations.
  • #1
freydawg56
22
0
I am currently studying a great text

Elementary Differential Equations and Boundary Valued Problems 9th edition;
and we have come to chapter 5 and are studying Ordinary Points, Singular Points, and Irregular Points. (get the point?)

Anyway, I did see these mentioned,,

this Bessel equation:

x^2 y`` + x y` + (x^2 - v^2) = 0

and the Legendre equation:
(1 - x^2) y`` -2x y` [tex]\alpha[/tex] ([tex]\alpha[/tex] + 1) = 0

and since they have their own names they do "seem" important.

This is just one chapter of my studies this semester but would anyone care to inform me of any physical relevance of these equations to the real world? I'm sure they help in some way, and I suppose I could google it, but this is the physics forums and I'm sure you guys love this stuff as much as I enjoy typing this out right now. so PF's help me please!
 
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  • #2
The Bessel and Legendre equations are what you get when you write [itex]\nabla^2 f(x, y, z)= 0[/itex] in cylindrical and spherical coordinates, respectively.

And that is important because the "Laplacian", [itex]\nabla^2 f[/itex] is the simplest second order differential operator in 3 dimensions that is "invariant under rigid motions"- that is, rotating or translating the coordinates does not change its form. And because of that, it shows up in many physical problems. For example:
potential energy: [itex]\nabla^2 f(x,y,z)= 0[/itex]
the wave equation: [itex]\nabla^2 f(x,y,z,t)= \frac{\partial^2 f}{\partial t^2}[/itex]
the heat or diffusion equation: [itex]\nabla^2 f(x,y,z,t)= \frac{\partial f}{\partial t}[/itex]
 

1. Why is it important to derive regular singular points?

Deriving regular singular points is important because it allows us to understand the behavior of a system at these specific points. This can help us make predictions and solve complex problems in various scientific fields, such as physics and engineering.

2. What are regular singular points?

Regular singular points are points in a system where the equations describing the system become singular (or undefined) but the solutions are still well-behaved. In other words, the system may have a singularity at these points, but it is not a pathological one.

3. What is the difference between regular and irregular singular points?

The main difference between regular and irregular singular points is that regular singular points have well-behaved solutions, whereas irregular singular points have solutions that are not well-behaved. This means that we can analyze and make predictions at regular singular points, but not at irregular ones.

4. How do we derive regular singular points?

There are several methods for deriving regular singular points, depending on the specific system and equations involved. Some common techniques include using power series solutions, Frobenius method, and transformation of variables.

5. Can regular singular points exist in all types of systems?

Yes, regular singular points can exist in a wide range of systems, including linear and non-linear systems, as well as in both continuous and discrete systems. However, the methods for deriving them may vary depending on the type of system.

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