Re: Re: [Mathematica 6] Integrate strange result
- To: mathgroup at smc.vnet.net
- Subject: [mg78447] Re: [mg78432] Re: [Mathematica 6] Integrate strange result
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Mon, 2 Jul 2007 06:45:38 -0400 (EDT)
- References: <200707011147.HAA15189@smc.vnet.net>
On 1 Jul 2007, at 20:47, Bhuvanesh wrote:
>> notice in the above, it says for k<-1 there is NO answer
>
> That's not what the If structure is saying. It's giving a result
> for k>=-1 and the third argument of the If statement is held
> (unevaluated). For a given result If[cond, res, int], if you want
> the result when the condition cond is satisfied, you can extract
> that from the If (by using a replacement rule or by using Refine);
> if you want the other part, you have to explicitly evaluate the
> held integral.
>
> In[1]:= f = (1 + k*Sin[a]^2)^(1/2);
>
> In[2]:= Assuming[Element[k, Reals], Integrate[f, {a, 0, 2*Pi}]] //
> InputForm
>
> Out[2]//InputForm=
> If[k >= -1, 4*EllipticE[-k], Integrate[Sqrt[1 + k*Sin[a]^2], {a, 0,
> 2*Pi},
> Assumptions -> k < -1]]
>
> In[3]:= PiecewiseExpand[%] //InputForm
>
> Out[3]//InputForm= Piecewise[{{4*EllipticE[-k], k >= -1}}, 0]
>
> Bhuvanesh,
> Wolfram Research
>
There is only one problem with this, already pointed out in this
thread. The last answer is wrong and is contradicted by Mathematica
itself:
k = -2;
f = (1 + k*Sin[a]^2)^(1/2);
Integrate[f, {a, 0, 2*Pi}]
4*EllipticE[2]
Note that this is the case k<-1, and the answer is not 0. Since 2
lies on the branch cut of EllipticE one might like to confirm this
numerically:
NIntegrate[f, {a, 0, 2*Pi}, AccuracyGoal -> 3]
2.396313190851802 + 2.396305377107839*I
which is pretty clearly non-zero and in good agreement with
N[4*EllipticE[2]]
2.396280469471184 + 2.396280469471184*I
Andrzej Kozlowski
- References:
- Re: [Mathematica 6] Integrate strange result
- From: Bhuvanesh <lalu_bhatt@yahoo.com>
- Re: [Mathematica 6] Integrate strange result