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Math Help - How to solve an integral.

  1. #1
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    How to solve an integral.

    How to solve the following integral (in Maple notation):

    Int(Int((exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u),v);

    I am aware that this integral can be written as:

    Int((exp(-z))/((1+exp(-z))^2),z);

    with z = u^2+v^2-2*rho*u*v

    I am especially interested in the following cases:

    Int(Int((exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u=-infinity..x),v=-infinity..y);

    Int(Int(u*v*(exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u=-infinity..infinity),v=-infinity..infinity);

    Thanks a lot for any help.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: How to solve an integral.

    Quote Originally Posted by Ad van der Ven View Post
    Int(Int((exp(-(u^2+v^2-2*rho*u*v)))/((1+exp(-(u^2+v^2-2*rho*u*v)))^2),u),v);
    It is rather difficult to understand it. Perhaps you mean \iint_{\mathbb{R}^2}\frac{e^{-(u^2+v^2-2\rho uv)}}{1+e^{-(u^2+v^2-2\rho uv)}}dudv . If so, and 1-\rho^2>0 the quadratic form Q(u,v)=u^2+v^2-2\rho uv=(u,v)\begin{pmatrix}{1}&{-\rho}\\{-\rho}&{1}\end{pmatrix}\begin{pmatrix}{u}\\{v}\end{  pmatrix} is positive definite and by means of the Spectral Theorem (diagonalizing Q) and the Euler's integral you can express the given integral in terms of product of the eigenvalues of the matrix of Q (i.e. its determinant 1-\rho^2) .
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