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Re: Problem with integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg123710] Re: Problem with integration
  • From: "Oleksandr Rasputinov" <oleksandr_rasputinov at hmamail.com>
  • Date: Fri, 16 Dec 2011 05:48:13 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jccgt4$mlv$1@smc.vnet.net>

On Thu, 15 Dec 2011 10:07:32 -0000, Mehdi Mortazawi  
<mehdimortazawi at gmail.com> wrote:

> Hi,
> I was curious if there is difference between online Mathematica and
> the one I have installed on my MAC(Mathematica version: 8.0.4.0).
> The question rises from the problem that I am facing with numerical
> integration. I want to integrate the following equation:
>
> NIntegrate[e^(-x) ((1 + 2 x)/(1)) (BesselK[0, 2 Sqrt[x] Sqrt[1+x]]
> +BesselK[2, 2 Sqrt[x] Sqrt[1 + x]]) + (4 x^2 +2 x-1)*BesselK[1, 2
> Sqrt[x] Sqrt[1+x]]*e^(-x)/((1)Sqrt[x] Sqrt[1 + x]),{x,0,1}]
>
> When I type the equation on my computer, I receive the following
> message:
> " ...has evaluated to non-numerical values for all sampling points in
> the region with boundaries {{0,1}"
> but the online version gives the following answer which is the value I
> am expecting:
>
> http://www.wolframalpha.com/input/?i=NIntegrate[e^%28-x%29+%28%281+%2B+2+x%29%2F%281%29%29+%28BesselK[0%2C+2+Sqrt[x]+Sqrt[1%2Bx]]%2BBesselK[2%2C+2+Sqrt[x]+Sqrt[1+%2B+x]]%29+%2B+%284+x^2+%2B2+x-1%29*BesselK[1%2C+2+Sqrt[x]+Sqrt[1%2Bx]]*e^%28-x%29%2F%28%281%29Sqrt[x]+Sqrt[1+%2B+x]%29%2C{x%2C0%2C1}]
>

The difference in this case is that Wolfram Alpha is more lenient with  
respect to what it accepts as input. The reason Mathematica doesn't give  
an answer is because e is not defined. But if you define it, say as  
\[ExponentialE], then you will get:

e = \[ExponentialE];
NIntegrate[
  e^(-x) ((1 + 2 x)/(1)) (
   BesselK[0, 2 Sqrt[x] Sqrt[1 + x]] +
    BesselK[2, 2 Sqrt[x] Sqrt[1 + x]]
   ) + (4 x^2 + 2 x - 1)*BesselK[1, 2 Sqrt[x] Sqrt[1 + x]]*e^(-x)/((1)  
Sqrt[x] Sqrt[1 + x]),
  {x, 0, 1}
]

1.245

Wolfram Alpha gives this answer after having guessed that the intended  
value of e is \[ExponentialE]. But if you want the same result from  
Mathematica, you have to be specific about the values involved.



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