Multiple integral

Let \langle \cdot, \cdot \rangle denote the usual inner product of \mathbb{R}^m. Evaluate the integral

    \[\mathcal{M} = \int \limits_{\mathbb{R}^m} \exp \left( - ( \langle x, \mathcal{S} ^{-1} x \rangle )^a \right) \, {\rm d}x\]

where \mathcal{S} is a positive symmetric m \times m matrix and a>0.

Solution

Since \mathcal{S} is a positive symmetric matrix , so is \mathcal{S}^{-1}. For a positive symmetric matrix \mathcal{A} there exists an \mathcal{R} positive symmetric matrix such that \mathcal{A} = \mathcal{R}^2. Applying this to \mathcal{S}^{-1} our integral becomes

    \[\mathcal{M} = \int \limits_{\mathbb{R}^m} \exp \left ( - \left \| \mathcal{R} x \right \|^{2a} \right ) \, {\rm d}x\]

where \left \| \cdot \right \| is the Euclidean norm. Applying a change of variables we have that

    \[\mathcal{M} = \det \left ( \mathcal{R}^{-1} \right ) \int \limits_{\mathbb{R}^m} e^{-\left \| y \right \|^{2a}} \, {\rm d}y\]

Since \det \left ( \mathcal{R}^{-1} \right ) = \sqrt{\det \left ( \mathcal{S} \right )} then by converting to polar coordinates we have that

    \begin{align*} \mathcal{M} &=  \omega_m \sqrt{\det \left ( \mathcal{S} \right )} \int_{0}^{\infty} r^{m-1} e^{-r^{2a}} \, {\rm d}r \\ &= \frac{\omega_m}{m} \sqrt{\det \left ( \mathcal{S} \right )}\Gamma \left ( \frac{m}{2a} + 1 \right ) \end{align*}

Here \omega_m denotes the surface area measure of the unit sphere and it is known to be

    \[\omega_m = \frac{2 \pi^{m/2}}{\Gamma \left ( \frac{m}{2} \right )}\]

hence

    \[\mathcal{M}=\frac{\sqrt{\det (\mathcal{S}) }\pi^{m/2}\Gamma\left(\frac{m}{2a}\right)}{2^{a-1}\Gamma\left(\frac{m}{2}\right)}\]

where \Gamma denotes the Gamma Euler function for which it holds that

    \[\Gamma(x+1) = x \Gamma(x) \quad \text{forall} \quad x>0\]

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