Confluent hypergeometric function: Difference between revisions

Revision as of 21:17, 7 January 2021

Revision as of 17:52, 23 January 2021

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::<math>M_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

::<math>W_{\kappa,\mu}(z) = e^{-\tfrac{z}{2}} z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+\tfrac{1}{2}, 1+2\mu; z\right)</math>

* The general {{mvar|p}}-th raw moment ({{mvar|p}} not necessarily an integer) can be expressed as<ref>{{CitationCite neededweb|datetitle=MarchAspects 2017of Multivariate Statistical Theory {{!}} Wiley|url=https://www.wiley.com/en-us/Aspects+of+Multivariate+Statistical+Theory-p-9780471769859|access-date=2021-01-23|website=Wiley.com|language=en-us}}</ref>

:: <math>\begin{align}

\operatorname{E} \left[\left|N\left(\mu, \sigma^2 \right)\right|^p \right] &= \frac{\left(2 \sigma^2\right)^{p/2} \Gamma\left(\tfrac{1+p}{2}\right)}{\sqrt \pi} \ {}_1F_1\left(-\tfrac p 2, \tfrac 1 2, -\tfrac{\mu^2}{2 \sigma^2}\right)\\