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19 Elliptic IntegralsApplications

§19.31 Probability Distributions

RG(x,y,z) and RF(x,y,z) occur as the expectation values, relative to a normal probability distribution in 2 or 3, of the square root or reciprocal square root of a quadratic form. More generally, let A (=[ar,s]) and B (=[br,s]) be real positive-definite matrices with n rows and n columns, and let λ1,,λn be the eigenvalues of AB-1. If x is a column vector with elements x1,x2,,xn and transpose xT, then

19.31.1 xTAx=r=1ns=1nar,sxrxs,

and

19.31.2 n(xTAx)μexp(-xTBx)dx1dxn=πn/2Γ(μ+12n)detBΓ(12n)Rμ(12,,12;λ1,,λn),
μ>-12n.

§19.16(iii) shows that for n=3 the incomplete cases of RF and RG occur when μ=-1/2 and μ=1/2, respectively, while their complete cases occur when n=2.

For (19.31.2) and generalizations see Carlson (1972b).