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Re: Mod Bessel function bug ?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg23027] Re: [mg22988] Mod Bessel function bug ?
*From*: "J.R. Chaffer" <jrchaff at nwlink.com>
*Date*: Tue, 11 Apr 2000 23:18:45 -0400 (EDT)
*References*: <b6.397c462.2623cd22@aol.com>
*Sender*: owner-wri-mathgroup at wolfram.com
Thanks for the reply. This may be where I am having confusion,
please permit me to continue so I can learn where I am going
wrong.
A book I have (Wylie & Barrett, "Adv. Eng. Mathematics", 6E),
says that a solution to the equation I am after can be expressed
as
c1*sqrt(x)*I{1/4,x^2} + c2*sqrt(x)*K{-1/4,x^2}; (p. 798)
It goes on to state that, quoting, "..if order is not an integer,...
K(order) may be replaced by.. I(-order), if desired".
for n not integer, I jumped to the conclusion that I(-n,x) = K(n,x)
for n not integer. I am thinking now that the fact that the general
solution can be expressed as EITHER c1*I(n,x) + c2*I(-n,x),
OR c3*I(n,x) + c4*K(n,x), does not mean that the two second
functions are identical. And in fact, J(-n,x) is not the same as Y(n,x),
but rather Y(n,x) is a linear combination of J(n,x) and J(-n,x).
Not the same thing.
Thank you for the insight.
J.R. Chaffer
-------
BobHanlon at aol.com wrote:
> Plot[Evaluate[Table[BesselK[n, x], {n, 0, 2, 1/3}]], {x, 0.1, 1.5}];
>
> Series[BesselK[n, x], {x, 0, 5}] == Series[BesselK[-n, x], {x, 0, 5}] //
> Normal // Simplify
>
> True
>
> FullSimplify[BesselK[n, x] == BesselK[-n, x]]
>
> True
>
> Abramowitz & Stegun 10.2.4 argues against your statement that "the I function
> with negative, noninteger order is the same as the K function"
>
> Sqrt[1/2*Pi/z]*BesselK[n + 1/2, z] ==
> Pi/2 * (-1)^(n + 1)*
> Sqrt[1/2*Pi/z]*(BesselI[n + 1/2, z] - BesselI[-n - 1/2, z]);
>
> Simplify[FunctionExpand[%], Element[n, Integers]]
>
> Sqrt[Pi/2]*Sqrt[1/z]*(-BesselK[-(1/2) - n, z] + BesselK[1/2 + n, z]) == 0
>
> FullSimplify[% /. n -> (m - 1/2)]
>
> True
>
> Bob Hanlon
>
> In a message dated 4/10/2000 10:16:07 AM, jrchaff at nwlink.com writes:
>
> >Well, something is very strange.
> >
> >Thank both of you for your replies. I am using Mathematica
> >Student Version 4.0; supposedly same as full version
> >capabilities, or at least so advertised.
> >
> >My plots show there is a difference near zero; however, both
> >functions come together (and become large) for large argument,
> >precisely opposite to what bessel theory says. Supposedly,
> >the I function with negative, noninteger order is the same as
> >the K function (not with Mathematica), and the K function
> >goes to zero exponentially with large argument.
> >
> >Something is definitely wrong.
> >
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