MultivariateGaussianComptonProfile¶

\[J(y) = \frac{1}{\sqrt{2\pi} \sigma_{x} \sigma_{y} \sigma_{z}} \frac{2}{\pi} \int_{0}^{1} d(cos \theta) \int_{0}^{\frac{\pi}{2}} d \phi S^{2}(\theta, \phi) exp \left( -\frac{y^{2}} {2 S^{2}(\theta, \phi)} \right)\]

\[\frac{1}{S^{2}(\theta, \phi)} = \frac{sin^{2}\theta cos^{2}\phi}{\sigma_{x}^{2}} + \frac{sin^{2}\theta sin^{2}\phi}{\sigma_{y}^{2}} + \frac{cos^{2}\theta}{\sigma_{z}^{2}}\]

\[-A_{3}(q)\frac{d^{3}}{dy^{3}}J(y) = \frac{\sigma_{x}^{4} + \sigma_{x}^{4} + \sigma_{x}^{4}} {9 \sqrt{2 \pi} \sigma_{x} \sigma_{y} \sigma_{z} q} \int_{0}^{1} d(cos \theta) \int_{0}^{\frac{\pi}{2}} d \phi \left[ \frac{y^{3}}{S^{2}(\theta, \phi)^{4}} -3 \frac{y}{S^{2}(\theta, \phi)^{2}} \right] S^{2}(\theta, \phi) exp \left( -\frac{y^{2}} {2 S^{2}(\theta, \phi)} \right)\]