Multivariable integral, probably related to the gamma functionLet x = [ x 1 , x 2 , . . , x n ] T represent the vector of all variables and D be a diagonal matrix, the question is to integrate or give an approximate answer: ∫ ⋯ ∫ [ 0 , ∞ ] n ( x T D x ) α − 1 e x p ( − x T D x ) d x 1 … d x n

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Multivariable integral, probably related to the gamma functionLet   x  =            [              x        1            ,              x        2            ,      .      .      ,              x        n            ]              T       represent the vector of all variables and D be a diagonal matrix, the question is to integrate or give an approximate answer:   ∫  ⋯      ∫          [      0      ,      ∞              ]                  n                      (      x          T        D  x      )          α      −      1        e  x  p  (  −      x          T        D  x  )    d      x    1    …  d      x    n

g5riem7z · Accepted Answer

The integral is easy, let me show you why:  ∫  ⋯      ∫          [      0      ,      ∞              ]                  n                      (      x          T        D  x      )          α      −      1        e  x  p  (  −      x          T        D  x  )    d      x    1    …  d      x    n  let&#039;s first express:      x    T    D  x  =      ∑          i      =      1        n        d    i          x    i    2  where       d    i    =      D          i      ,      i      To have a convergent integral every di must be non negative. hence we can use an alternate coordinate system:      η    i    =            d      i            x    i  and       x    T    D  x becomes:      x    T    D  x  =      ∑          i      =      1        n        d    i          x    i    2    =      ∑          i      =      1        N        η    i    2    =      η    T    ηand the integral using substitution obviously becomes:      I    k    ≡  ∫  ⋯      ∫          [      0      ,      ∞              ]                  n                      (      η    T    η      )    k    e  x  p  (  −      η    T    η  )              d              η        1            …      d              η        n                            ∏                  i          =          1                n                    d        i            Now look at this technique : Differentiation by a parameter  T  (  β  )  ≡  ∫  ⋯      ∫          [      0      ,      ∞              ]                  n                      e  x  p  (  −  β      η    T    η  )    d      η    1    …  d      η    n              ∂      k                      ∂        β            k        T  (  β  )  =  ∫  ⋯      ∫          [      0      ,      ∞              ]                  n                      (  −      η    T    η      )    k    e  x  p  (  −  β      η    T    η  )    d      η    1    …  d      η    n  so      I    k    =            (      −      1              )        k                            ∏                  i          =          1                n                    d        i                  lim          β      →      1                  ∂      k                      ∂        β            k        T  (  β  )now all that there is to it is calculating   T  (  β  )  T  (  β  )  ≡  ∫  ⋯      ∫          [      0      ,      ∞              ]                  n                      e  x  p  (  −  β      η    T    η  )    d      η    1    …  d      η    n    =            (              ∫        0                  ∞                    e      x      p      (      −      β              z        2            )            d      z      )        n    =            (              1        2                              π          β                    )        n

Multivariable integral, probably related to the gamma function Let x=[x_1,x_2,..,x_n]^{T}represent the vector of all variables and D be a diagonal matrix, the question is to integrate or give an approximate answer: idotsint_{[0,infty]^{n}} (x^{T}Dx)^{alpha-1}exp(-x^{T}Dx) ,dx_1 dots dx_n

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