5
$\begingroup$

I want to calculate the Bessel function, given by

$$J_\alpha (\beta) = \sum_{m=0}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha +1)} \left(\frac{\beta}{2}\right)^{2m}$$

I know there are some tables that exist for this, but I want to keep the $\beta$ variable (i.e. I want a symbolic form in terms of $\beta$). If there is a way to simplify the summation part of the equation and leave an equation only in terms of $\beta$, that would be very helpful. (I see there is a dependence on $2m$, but I would like to see a way to break down the "other half" of the equation.)

Another question I have is: how is this calculated for $\beta$ values that are greater than $1$? It seems to me that this would give an infinite sum.

I am looking for something for $\alpha=1,3,5$ and $\beta=4$.

Thanks in advance.

  • 0
    Perhaps you would be interested in [an integral representation for the Bessel function](http://dlmf.nist.gov/10.9)?2012-05-06
  • 0
    thanks for fixing up my LaTex code, you beat me to it! Would the integral representation get rid of the infinite sum? What i want to do is to get a function just in terms of $\beta$ so that I can manipulate it. preferably the sigma summation can be simplified to a constant or an approximation.2012-05-06
  • 0
    Sure thing! Any of those integral representations are equal to the sum you gave (where applicable, look to the right of the formulas on that page to see their domains). If you're interested in an approximation, say for large $\beta$, you may take as many or as few terms as you like from [an asymptotic expansion for the Bessel function](http://dlmf.nist.gov/10.17). I strongly urge you to take a look around [the rest of that site's Bessel function category](http://dlmf.nist.gov/10).2012-05-06
  • 2
    The best algorithm to use very much depends on what kind of parameters you are interested in. For $\alpha$: is it an integer, or an arbitrary real number, or an arbitrary complex number? For $\beta$, is it small or large? To give an example: there are what are called *asymptotic expansions* you can use for $\alpha$ large, $\beta$ large, or both $\alpha$ and $\beta$ large. For other arguments, things are not too simple.2012-05-06
  • 0
    Antonio, at first glance, it seems that Eq. 10.17.3 gives even more complicated equations than before. To answer J.M.'s question (and maybe for a little more detail to my problem), I am looking for something when $\alpha$ values of 1, 3, 5 and $\beta$ equal to 4. This is troubling as I cannot really use any of the limiting formulas referred to by Antonio. Please advise.2012-05-06
  • 0
    @J.M. just pinging you for suzu's comment above.2012-05-06
  • 0
    For $\beta > 1$ ... use the standard formula, and compute the radius of convergence of this power series. The result: it converges for all $\beta$, including $\beta > 1$, and even for complex $\beta$, matrix $\beta$, and other exotic things.2012-05-10

4 Answers 4