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Thread: 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 $\displaystyle \iint_{\mathbb{R}^2}\frac{e^{-(u^2+v^2-2\rho uv)}}{1+e^{-(u^2+v^2-2\rho uv)}}dudv$ . If so, and $\displaystyle 1-\rho^2>0$ the quadratic form $\displaystyle Q(u,v)=u^2+v^2-2\rho uv=(u,v)\begin{pmatrix}{1}&{-\rho}\\{-\rho}&{1}\end{pmatrix}\begin{pmatrix}{u}\\{v}\end{ pmatrix}$ is positive definite and by means of the Spectral Theorem (diagonalizing $\displaystyle Q$) and the Euler's integral you can express the given integral in terms of product of the eigenvalues of the matrix of $\displaystyle Q$ (i.e. its determinant $\displaystyle 1-\rho^2$) .
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