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MathGroup Archive 2007

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Re: Simplifying expressions containing Bessel functions?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg76864] Re: Simplifying expressions containing Bessel functions?
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Tue, 29 May 2007 04:50:37 -0400 (EDT)
  • References: <f3dpbp$ft0$1@smc.vnet.net>

FunctionExpand and FullSimplify know rules like this.

In[41]:=
o = D[x*BesselJ[1, x], x]
Through[{FunctionExpand, FullSimplify}[o]]

Out[41]=
BesselJ[1, x] + (1/2)*x*(BesselJ[0, x] - BesselJ[2, x])
Out[42]=
{x*BesselJ[0, x], x*BesselJ[0, x]}

However you should use FunctionExpand for issues like this.

As other examples consider

In[49]:=
D[x^2*BesselJ[10, x], x, x, x]
Through[{FunctionExpand, FullSimplify}[%]]
Out[49]=
BesselJ[9, x] + 2*(BesselJ[9, x] - BesselJ[11, x]) - BesselJ[11, x] +
  3*x*((1/2)*(BesselJ[8, x] - BesselJ[10, x]) + (1/2)*(-BesselJ[10, x]
+ BesselJ[12, x])) +
  (1/2)*x^2*((1/2)*((1/2)*(BesselJ[7, x] - BesselJ[9, x]) + (1/2)*(-
BesselJ[9, x] + BesselJ[11, x])) +
    (1/2)*((1/2)*(-BesselJ[9, x] + BesselJ[11, x]) +
(1/2)*(BesselJ[11, x] - BesselJ[13, x])))

Out[50]=
{((12960 - 192*x^2 + x^4)*BesselJ[7, x])/x^2 - (3*(69120 - 1264*x^2 +
7*x^4)*BesselJ[8, x])/x^3,
  (x*(12960 - 192*x^2 + x^4)*BesselJ[7, x] - 3*(69120 - 1264*x^2 +
7*x^4)*BesselJ[8, x])/x^3}

In[59]:=
Integrate[x*BesselJ[2, x], x, x, x]
Through[{FunctionExpand, FullSimplify}[%]]

Out[59]=
(-x^2)*HypergeometricPFQ[{1/2}, {3/2, 2}, -(x^2/4)] -
(1/24)*x^4*HypergeometricPFQ[{3/2}, {5/2, 3}, -(x^2/4)]
Out[60]=
{(1/2)*BesselJ[1, x]*(8*x - 3*Pi*x^2*StruveH[0, x]) + (3/2)*BesselJ[0,
x]*(-2*x^2 + Pi*x^2*StruveH[1, x]),
  (1/2)*x*(BesselJ[1, x]*(8 - 3*Pi*x*StruveH[0, x]) + 3*x*BesselJ[0,
x]*(-2 + Pi*StruveH[1, x]))}

BTW, recently it was a thread about BeselJ. Take a look here

http://groups.google.gr/group/comp.soft-sys.math.mathematica/browse_thread/thread/4f43efab17e4515d/74f2e76fe93c9e21?lnk=gst&q=BesselJ&rnum=1&hl=el#74f2e76fe93c9e21

Dimitris


 /  AES       :
> Is there any kind of Simplification procedure in Mathematica that will
> somehow apply the recursion relations between Bessel functions of orders
> n-1, n and n+1 to eliminate the highest orders in an expression?
>
> I realize this is not a simple topic -- but it would be nice if
>
>    D[x BesselJ[1, x], x]
>
> would yield
>
>    x BesselJ[0, x]
>
> rather than
>
>    BesselJ[1, x]  +  (x/2) ( BesselJ[0, x] - BesselJ[2, x] )



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