Re: Solving double integral without Error function

• To: mathgroup at smc.vnet.net
• Subject: [mg57724] Re: Solving double integral without Error function
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Mon, 6 Jun 2005 04:21:33 -0400 (EDT)
• Organization: The University of Western Australia
• References: <d7rk3i\$bkt\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <d7rk3i\$bkt\$1 at smc.vnet.net>,
"Xavier Brusset" <brusset at poms.ucl.ac.be> wrote:

> I have to solve this double integral:
> Phi[ q_ ]=Integrate[Integrate[y(x-q)f[x,y],{x,0,q}],{y,2,15}]
> where
> f[ x_ , y_ ]=1/(2 Pi Sqrt[27])Exp{-1/54 [4 x^2 - 6yx - 50x + 9y^2 -30 y +
> 325]}

There is a syntax error here in your definition of f.

> Mathematica 5.1 on windows fumbles a lot and finally stops giving an answer
> in q but using the Error function (I am using the Statistics Continuous
> distributions module).

That is irrelevant. Integrating exponentials of quadratic functions will
lead to Erf. In general, there is no simple closed form for a second
(indefinite) integral.

> The problem with the Error function is that when I combine this function Phi
> into another equation, I can't find a root in q.
> Which function of Mathematica should I use which circumvents the Error
> function in solving the double integral?
> Or, alternately, which root finding function could I use to find even a
> numeric instance of q?

There is a simple alternative way of proceeding. It is straightforward
to compute the first and second derivatives of Phi with respect to q.
You can use this with NDSolve to compute Phi[q]. See

http://physics.uwa.edu.au/pub/Mathematica/MathGroup/ErfIntegral.nb

Cheers,
Paul

--
Paul Abbott                                      Phone: +61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)
AUSTRALIA                               http://physics.uwa.edu.au/~paul
http://InternationalMathematicaSymposium.org/IMS2005/

```

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