Thoughts on Bessel functions
- From: "Gene Ward Smith" <genewardsmith@xxxxxxxxx>
- Date: 4 Jul 2006 17:42:38 -0700
Consider the homogenous differential equation
x y'' + (n+1) y' = y
where n is a fixed complex number. If n is not a negative integer, we
can develop one solution in series as
B_n(x) = \sum_i=0^infinity 1/(n+1)^(i) x^i/i! = \sum_i=0^infinity
x^i/(C(n+i,n) i!^2)
Here a^(b) = Gamma(a+b)/Gamma(a) is the Pochhammer symbol, and C(a,b)
the extended binomial coefficient function.
The Bessel functions and modified Bessel functions of the first kind
can both be expressed in terms of B_n. We have
B_n(x) = n! I_n(2 sqrt(x))/x^(n/2)
so that
I_n(z) = 1/n! (z/2)^n B_n(z^2/4)
Similarly, we have
J_n(z) = 1/n! (z/2)^n B_n(-z^2/4)
Outside of n equaling a negative integer, this function B is better
behaved than the Bessel functions, since it is entire. It has a simpler
differential equation. It expresses both the J and I functions in terms
of a single function without requiring imaginary arguments. Arguably,
it seems to me, it is a better way to start out the theory of Bessel
functions. What I'm wondering is if anyone has actually done this, and
if B_n has a name. Also, can anyone see why this approach isn't used,
if in fact it is not?
.
- Follow-Ups:
- Re: Thoughts on Bessel functions
- From: david petry
- Re: Thoughts on Bessel functions
- Prev by Date: Re: Recognize this equation ?
- Next by Date: Re: New high precision scientific calculator
- Previous by thread: help: a question on limit
- Next by thread: Re: Thoughts on Bessel functions
- Index(es):
Relevant Pages
|
