Re: Re: Jacobi polynomials in Mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg99083] Re: [mg99063] Re: Jacobi polynomials in Mathematica
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Sun, 26 Apr 2009 01:40:07 -0400 (EDT)
- References: <gsrqsm$rac$1@smc.vnet.net> <gss3d4$2po$1@smc.vnet.net>
- Reply-to: drmajorbob at bigfoot.com
This is entirely correct:
Product[j, {j, 1, 4.5}]
24
...since j varies from the lower limit (1 in this case), adding 1 at each
step, and stopping at OR BELOW the upper limit. In this case j takes on
the values 1, 2, 3, and 4. Adding 1 again to get 5 would exceed the upper
limit.
But consider the calculations
Product[j, {j, 1, n}]
% /. n -> 4.5
Gamma[4.5 + 1]
n!
52.3428
52.3428
The first result generalizes the Product to an arbitrary positive integer
and returns Factorial, an INCORRECT result for non-integers such as 4.5.
Yet it's a USEFUL result for non-integers, if we intend to use the Gamma
function to extend Factorial.
Bobby
On Sat, 25 Apr 2009 03:52:46 -0500, Cora L <cora.lahnstein at googlemail.com>
wrote:
> On 24 Apr, 11:13, Jens-Peer Kuska <ku... at informatik.uni-leipzig.de>
> wrote:
>> Hi,
>>
>> a simple check
>>
>> JacobiP[n, a, b, z] /. {n -> 0.5, a -> 0.5, b -> 0.5, z -> 0.2}
>>
>> gives
>>
>> 0.767081
>>
>> so, it may be defined.
>>
>> Regards
>> Jens
>>
>> Cora L wrote:
>> > Hello,
>> > I have a simple question: in Mathematica the Jacobi polynomials are
>> > implemented
>> > as JacobiP[n, a, b, z], see
>> >http://mathworld.wolfram.com/JacobiPolynomial.html
>>
>> > Is JacobiP[n, a, b, z] also defined if n is not an integer? More
>> > general, is
>> > JacobiP[n, a, b, z] defined for all real n, a, b and z?
>>
>> > Thanks!
>
> Well, even if Mathematica is giving out a value I'm not too sure
> whether it's correct or not.
>
> For example,
> Product[j, {j, 1, 4}] gives 24
>
> But
> Product[j, {j, 1, 4.5}] also gives 24.
>
> Surely, the second answer is wrong.
>
--
DrMajorBob at bigfoot.com