Bessel's equation

[MaxFusion16]

hi
i'm ashamed to ask for math help for i know you will mock me, but whatever. So i have this calc3 project, and the question is can a polynomial solve bessel's equation, and if so which polynomial. Is an infinite series considered a polynomial? any help would be greatly appreciated.

 

[miniMUNCH]

Well,

A potential solution to Bessels equation is a linear combination of linearly independent infinite series expansions about a regular singular point using the method of Frobenius. These two linearly independent infinite series are better known as Bessel functions [of various kinds and orders].

I suppose you could truncate the Bessel function solutions after some large but finite amount of terms that would technically be a polynomial albeit an extremely large one --> but this would only yield an approximate solution that would only be valid some finite radius (in the real and complex plane) from the origin. The magnitude of the origin would depend upon the number of terms from the infinite series used.

In some cases, if you expanded about the origin for instance, yon might need hundreds of thousands of terms just to represent the function to a radius of 10 or so (i.e. x= + or - 10)!

Hope that helps!

 

[RaynorWolfcastle]

Mathworld knows all

I remember that Bessel functions were related to frobenius, but I purged that part of my diff eqs class from memory because I hated solving those equations too much. Anyway, have fun with the above link 😉