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Thoughts on Bessel functions
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Wed Jul 05, 2006 12:19 am    Post subject: Thoughts on Bessel functions Reply with quote

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?
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Richard Mathar
science forum beginner


Joined: 23 May 2005
Posts: 45

PostPosted: Fri Jul 07, 2006 6:41 pm    Post subject: Re: Thoughts on Bessel functions Reply with quote

In article <1152058753.163472.270600@a14g2000cwb.googlegroups.com>,
genewardsmith@gmail.com writes:
Quote:
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?


The B_n(z^2/4) is usually written (see 9.6.47 in the Handbook of
Mathematical Functions, edited by Abramowitz and Stegun) as 0F1(n+1,z^2/4),
so it is too close to a special case of hypergeometric and confluent
hypergeometric functions to spend a new name on it.

The use of Bessel functions is first of all tied to the fact that they
appear as solutions to many differential equations in engineering and physics;
so everybody would rather use the "old" Bessel function notation instead
of any "new" B-Function notation if the solution to such a real-world problem
is needed as a paper-saving argument. Also: simplicity of the differential
equation is not a unique argument: if we use diffraction theory of circular
pupils in optics, we need the Fourier transforms of Chebyshev polynomials and
immediately face the integral representations of the Bessel functions, so
simplicity of integral representations is also a nice feature.

Also: what would be the linearly independent counterpart of B_n? This must
be of equivalent simplicity to convince us to switch so we can patch
solutions to the DE across boundaries Smile
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Gene Ward Smith
science forum Guru


Joined: 08 Jul 2005
Posts: 409

PostPosted: Sat Jul 08, 2006 7:50 am    Post subject: Re: Thoughts on Bessel functions Reply with quote

Richard Mathar wrote:

Quote:
The B_n(z^2/4) is usually written (see 9.6.47 in the Handbook of
Mathematical Functions, edited by Abramowitz and Stegun) as 0F1(n+1,z^2/4),
so it is too close to a special case of hypergeometric and confluent
hypergeometric functions to spend a new name on it.

I found by consulting Watson that A. George Greenhill called it the
Bessel-Clifford function, however.

Quote:
The use of Bessel functions is first of all tied to the fact that they
appear as solutions to many differential equations in engineering and physics;
so everybody would rather use the "old" Bessel function notation instead
of any "new" B-Function notation if the solution to such a real-world problem
is needed as a paper-saving argument.

Greenhill also pointed out that sometimes the "new" Bessel functions
(which are actually older than Bessel, as Lagrange used them) sometimes
work better in some engineering situations.

Also: simplicity of the differential
Quote:
equation is not a unique argument: if we use diffraction theory of circular
pupils in optics, we need the Fourier transforms of Chebyshev polynomials and
immediately face the integral representations of the Bessel functions, so
simplicity of integral representations is also a nice feature.

Once again, sometimes these become simpler, and sometimes they are not.

Quote:
Also: what would be the linearly independent counterpart of B_n? This must
be of equivalent simplicity to convince us to switch so we can patch
solutions to the DE across boundaries Smile

Here's an integral representation for it:

x^(-n/2) K_n(2 sqrt(x)) = (1/2) int_0^infinity exp(-t-x/t) dt/t^(n+1)

defined for Re(x) > 0.

This not only defines the other solution, x^(-n/2)K_n(2 sqrt(x)), it
provides an example of when the integral representation is simpler in
the "new" form.

If we don't normalize the hypergeometric way, but instead use
x^(-n/2)I_n(2 sqrt(x)), we get another very nice feature, by the way: a
function with which is holomorphic on C^2; in other words, in both
variables n and x.
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