Re: Simplifying Bessel functions

• To: mathgroup at smc.vnet.net
• Subject: [mg107287] Re: Simplifying Bessel functions
• From: Roland Franzius <roland.franzius at uos.de>
• Date: Mon, 8 Feb 2010 03:34:27 -0500 (EST)
• References: <hkj90t\$du1\$1@smc.vnet.net>

```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)

Out: {BesselJ[n, z] ->
(-z BesselJ[-2 + n, z] + 2 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.

--

Roland Franzius

```

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