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Re: solution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg118281] Re: solution
  • From: Peter Pein <petsie at dordos.net>
  • Date: Wed, 20 Apr 2011 04:27:07 -0400 (EDT)
  • References: <iojpp7$i52$1@smc.vnet.net>

Am 19.04.2011 12:56, schrieb amelia Jackson:
> Dear MathGroup,
>
> I have a problem. I want to find solution:
> r := Table[
> k /. FindRoot[BesselJ[0, k] + k BesselJ[1, k] == 0, {k, n}], {n, 1, 100}]
>
> but I get about 30 roots. I need about 100 or more.
> I think that "step" "n" tend to Pi
>
> Please for help...

Indeed the roots are asyptotically at (k+1/4) Pi for k->Infinity:


Limit[Abs[f[Pi*(k + 1/4)]] Sqrt[k], k -> Infinity,
    Assumptions -> Element[k, Integers]]

11/(4 Sqrt[2] \[Pi])

so Abs[f[Pi*(k + 1/4)]] ~ k^(-1/2)

leading to the much easier table of roots:

In[118]:= Length[
    zeros=Union[Table[x/. FindRoot[f[x],{x,(k+1/4) Pi}], {k,100}],
      SameTest->(Abs[#1-#2]<=1&)]]
Take[zeros,-3]

Out[118]= 100
Out[119]= {308.657,311.799,314.94}



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