#4325 Maxima can't differentiate beta function

[Raymond Toy]

An example:

(%i3) diff(beta(m,x),x);
                                d
(%o3)                           ── (beta(m, x))
                                dx

But if we convert the beta function to gamma functions, we get a derivative:

(%i7) makegamma(beta(m,x));
                               gamma(m) gamma(x)
(%o7)                          ─────────────────
                                 gamma(x + m)
(%i8) diff(%,x);
           gamma(m) psi (x) gamma(x)   gamma(m) gamma(x) psi (x + m)
                       0                                    0
(%o8)      ───────────────────────── - ─────────────────────────────
                 gamma(x + m)                  gamma(x + m)

This could be rewritten in terms of beta and psi[0]. There doesn't seem to be a function to convert gamma to beta functions.

[David Billinghurst]

The following works. I am not familiar with MQAPPLY and would welcome guidance on style.

;; Derivative of beta function
;; https://functions.wolfram.com/GammaBetaErf/Beta/
;; https://en.wikipedia.org/wiki/Beta_function

(defprop %beta
  ((a b)
    ; derivative wrt a
    ((mtimes)
      (($beta) a b)
      ((mplus)
        ((mqapply) (($psi array) 0) a)
        ((mtimes) -1
          ((mqapply) (($psi array) 0) ((mplus) a b)))))
     ; derivative wrt b
     ((mtimes)
      (($beta) a b)
      ((mplus)
        ((mqapply) (($psi) 0) b)
        ((mtimes) -1
          ((mqapply) (($psi array) 0) ((mplus) a b))))))
  grad)

(%i4) diff(beta(a,b),a);
(%o4) -(beta(a,b)*(psi[0](b+a)-psi[0](a)))
(%i5) diff(beta(a,b),b);
(%o5) -(beta(a,b)*(psi[0](b+a)-psi[0](b)))
(%i6) diff(beta(a,b),a,2);
(%o6) beta(a,b)*(psi[0](b+a)-psi[0](a))^2-beta(a,b)*(psi[1](b+a)-psi[1](a))
(%i7) diff(beta(a,b),b,2);
(%o7) beta(a,b)*(psi[0](b+a)-psi[0](b))^2-beta(a,b)*(psi[1](b+a)-psi[1](b))
(%i8) diff(beta(a,b),a,1,b,1);
(%o8) beta(a,b)*(psi[0](b+a)-psi[0](a))*(psi[0](b+a)-psi[0](b))
 -beta(a,b)*psi[1](b+a)

[Raymond Toy]

I think this is fine. Something similar is used in derivative of bessel_j. It seems that mqapply is how things like psi[0](x) are represented.

[David Billinghurst]

Fixed with commit [1be377]