What is ∂ ∂ x i ( x i ! ) where x i is a discrete variable?

Question

What is             ∂              ∂                  x          i                      (      x    i    !  ) where       x    i   is a discrete variable?

Jesse Reilly · Accepted Answer

It&#039;s probably best to use an analytic continuation of the factorial function, rather than the factorial itself. Consider the gamma function:  Γ  (  x  )  =      ∫          0              ∞            t          x      −      1            e          −      t        d  tObviously,   Γ  (  1  )  =  1, and we also have:                    Γ        (        x        +        1        )                            =                  ∫                      0                                ∞                                    t                      x                                    e                      −            t                          d        t                                          =        [        −                  t                      x                                    e                      −            t                                    ]                      0                                ∞                          +        x                  ∫                      0                                ∞                                    t                      x            −            1                                    e                      −            t                          d        t                                          =        x        Γ        (        x        )        .            So,   Γ  (  x  )  =  (  x  −  1  )  !. So, just freely take derivatives now.

What is d/(dx_i)(x_i!) where x_i is a discrete variable?

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New Questions in Calculus 1