Re: Simplifying Bessel functions

• To: mathgroup at smc.vnet.net
• Subject: [mg107310] Re: Simplifying Bessel functions
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Tue, 9 Feb 2010 02:42:57 -0500 (EST)
• References: <hkj90t\$du1\$1@smc.vnet.net> <hkoia7\$t02\$1@smc.vnet.net>

```Roland Franzius schrieb:
> Sam Takoy schrieb:
>> Hi,
>>
>> Mathematica does not seem to simplify the following expression:
>>
>> (BesselJ[2, BesselJZero[0, n]] BesselJZero[0, n]^2)/ BesselJ[1,
>>   BesselJZero[0, n]]
>>
>> (I believe the answer is 2 BesselJZero[0, n]^2)
>>
>> Is there a way of making Mathematica deal with these types of expressions?
>
> You want to Simplify
>
> In: expr = (BesselJ[2, BesselJZero[0, n]] BesselJZero[0, n]^2)/
>     BesselJ[1, BesselJZero[0, n]] /. BesselJZero[0, n] -> z
>
> Out:	(z^2 BesselJ[2, z])/BesselJ[1, z]
>
> Now Solve from the recursion formula for the highest order Bessel function
>
> In: rp[n_] = (Solve[
>      z BesselJ[n - 2, z] + z BesselJ[n, z] == 2 n BesselJ[n - 1, z] ,
>      BesselJ[n, z]] // First)

Sorry, has to be

rp[n_] = (Solve[z BesselJ[n - 2, z] + z BesselJ[n, z] ==
2 (n - 1) BesselJ[n - 1, z], BesselJ[n, z]] // First) // Apart

>
> Out: {BesselJ[n, z] ->
> 	(-z BesselJ[-2 + n, z] + 2 n BesselJ[-1 + n, z])/z}

{BesselJ[n, z] -> -BesselJ[-2 + n, z] +
(2 (-1 + n) BesselJ[-1 + n, z])/z}

>
> In: expr1 = Simplify[expr /. rp[2]]
>
> Out: z (4 - (z BesselJ[0, z])/BesselJ[1, z])
>
> In: expr2 = FullSimplify[expr1 /. z -> BesselJZero[0, n]]
>
> Out:
> BesselJZero[0, n] *
> 	(4 - (BesselJ[0, BesselJZero[0, n]] BesselJZero[0, n])/
>     		BesselJ[1, BesselJZero[0, n]])
>
> By definition
>
> BesselJ[0, BesselJ[0, n]] -> 0
>
> the result is
>
> In: expr2 = FullSimplify[expr1 /. z -> BesselJZero[0, n]]
>
> Out:
> BesselJZero[0, n] *
> 	(4 - (BesselJ[0, BesselJZero[0, n]] BesselJZero[0, n])/
>     	BesselJ[1, BesselJZero[0, n]])
>
> Mathematica does not reduce symbolic orders of Bessel functions.
>
> Assuming[ n > 0 && n \[Element] Integers, FullSimplify[expr2]]
>
>
> But for explicit integer n_ eg
>
> In: expr2 /. n -> 3
>
> Out: 4 BesselJZero[0, 3]
>
> it works.
>
> Hope it helps.
>
>

The correct result is

2 BesselJZero[0, n]

--

Roland Franzius

```

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