# Talk:Bessel function

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## Closure equation

I'm wondering how one would go about proving the closure equation that is cited in the article:

${\displaystyle \int _{0}^{\infty }xJ_{\alpha }(ux)J_{\alpha }(vx)dx={\frac {1}{u}}\delta (u-v)}$

Does anyone know how, or can they give me a reference? Thanks --Lavaka 05:35, 13 April 2007 (UTC)

I seem to recall that equation being from the Bowman book; perhaps it is proved there. —Steven G. Johnson 16:55, 13 April 2007 (UTC)
It's also found in Arfken and weber. They write: This may be proved by the use of Hankel transforms in section 15.1. An alternate approach, starting from a relation similar to (9.82), is given by Morse and Feshbach (Section 6.3). (9.82 is the equation for the Green's function of the Helmholtz equation.) —Steven G. Johnson 16:59, 13 April 2007 (UTC)
Thanks Steven, I'll try those --Lavaka 17:26, 13 April 2007 (UTC)

It's not in Bowman, but I found the same passage you did in Arfken and Weber, which isn't especially enlightening. I believe that the Hankel Transform, with respect to some variable ${\displaystyle u}$, of ${\displaystyle J_{0}(vx)}$ is indeed ${\displaystyle {\frac {1}{u}}\delta (u-v)}$, but I'm a bit unsure if the integral formulation is true. When we extend something like the Fourier Transform to include tempered distributions, no longer can we write certain integral formulations, even if the objects have a Fourier Transform. I don't know if Arfken and Weber are correct, but I haven't been able to find any other good sources. --Lavaka 20:56, 13 April 2007 (UTC)

PuZHANG (talk) 11:49, 14 November 2010 (UTC) I saw the passage in Arfken and Weber. They say people can prove the closure relation with Hankel transform. However, the Hankel transform pair is again based on this closure relation according to the item Hankel Transform in wikipedia. Actually to prove this relation is part of a problem in Jackson's Classical Electrodynamics (problem 3.16 in the 3rd ed.), where some hint is given. The proof should not be difficult following the hint.

What does this integral mean? Surely the integrand is not Lebesgue integrable because the integrand would have oscillations of approximately constant size out to infinity! Or more precisely, the integrand is like the product of two sine waves of different frequencies. It wouldn't even be a definable improper integral – the integral out to some X would have oscillations as a function of X that do not decay to zero amplitude. Eric Kvaalen (talk) 13:22, 29 July 2014 (UTC)

I have figured out the answer to my own question. If one defines a boxcar function of x that depends on a small parameter ε as:

${\displaystyle f_{\epsilon }(x)=\epsilon \ \mathrm {rect} \left({\frac {x-1}{\epsilon }}\right)}$

(where rect() is the rectangle function) then the Hankel transform of it (of any given order α), gε(k), approaches Jα(k) as ε approaches zero, for any given k. Conversely, the Hankel transform (of the same order) of gε(k) is fε(x):

${\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)g_{\epsilon }(k)dk=f_{\epsilon }(x)}$

which is zero everywhere except near 1. (This can be proved, at least for α zero, from Fourier analysis of a circularly symmetrical function, so it does not require the "closure" equation that we are discussing.) As ε approaches zero, this integral approaches δ(x−1). So by abuse of language (or "formally"), one might say that

${\displaystyle \int _{0}^{\infty }kJ_{\alpha }(kx)J_{\alpha }(k)dk=\delta (x-1)}$

even though the integral is not actually defined. A change of variables then yields the formula in question. Eric Kvaalen (talk) 09:23, 11 August 2014 (UTC)

## Integral representation of Bessel functions of the second kind

How to derive the Integral representation of Bessel functions of the second kind from its definition Y(x)={Jn(x)cos(n times pi)-J-n(x)}/sin(n times pi) with n tends to a integer ? I eager to know the proof because the Integral representation explain the asymptotic behaviour of Y with large x. —Preceding unsigned comment added by 61.18.170.29 (talkcontribs)

## Math symbols in section headings

The math symbols in the section headings — are they really needed? I removed them because I think it looks a little bit cluttery, and it makes it difficult to link to subsections from other articles since the math symbols are not plain text, although I noticed that there were redirection pages in order to more easily be able to link to those sections (but it took me a while to realize they existed). I also think the math symbols feel superfluous since they are mentioned in the body text. Or what is the reason they are written in the headings? —Kri (talk) 15:07, 26 February 2015 (UTC)

Many people navigate Bessel and Bessel-related functions by their symbol. This is not true of many functions, but certainly applies here. Limit-theorem (talk) 15:35, 26 February 2015 (UTC)
Okay. —Kri (talk) 15:45, 26 February 2015 (UTC)

## Always a Taylor expansion?

For non-integer order, the powers are non-integral and thus the series for the Bessel function of the first kind isn't a power series, but a Puiseux series or even a Hahn series. Even for integer order, negative orders would make it a Laurent series. Should we change the article to reflect this? It does still hold though that it is the product of ${\displaystyle x^{\alpha }}$ with a Taylor series.--Jasper Deng (talk) 07:12, 5 March 2015 (UTC)

## Confusing introduction to the topic about alpha

In the introduction, the second sentence is very unclear to me: "Although α and −α produce the same differential equation for real α, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.". The first thing I noticed is that alpha and -alpha produce the same differential equations for all alpha, not just real. The second part of that sentence is very vague/unclear. I hope someone can transform this sentence into something clearer. MicroVirus (talk) 10:24, 8 September 2015 (UTC)