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Beta_incomplete_regularized Calculator

Beta_incomplete_regularized

Function: beta_incomplete_regularized (<a>, <b>, <z>) The regularized incomplete beta function A&S 6.6.2, defined as beta_incomplete(a,b,z)/beta(a,b).

As for beta_incomplete this definition is not complete. See functions.wolfram.com for a complete definition of beta_incomplete_regularized.

beta_incomplete_regularized simplifies <a> or <b> a positive integer.

For z=0 and realpart(a)>0, beta_incomplete_regularized has the specific value 0. For <z=1> and realpart(b)>0, beta_incomplete_regularized simplifies to 1.

Maxima can evaluate beta_incomplete_regularized for real and complex arguments in float and bigfloat precision.

When beta_expand is true, Maxima expands beta_incomplete_regularized for arguments a+n or a-n, where n is an integer.

Maxima knows the derivatives of beta_incomplete_regularized with respect to the variables <a>, <b>, and <z> and the integral with respect to the variable <z>.

Examples:

Simplification for <a> or <b> a positive integer:

          (%i1) beta_incomplete_regularized(2,b,z);
                                                 b
          (%o1)                       1 - (1 - z)  (b z + 1)

          (%i2) beta_incomplete_regularized(a,2,z);
                                                         a
          (%o2)                         (a (1 - z) + 1) z

          (%i3) beta_incomplete_regularized(3,2,z);
                                                         3
          (%o3)                         (3 (1 - z) + 1) z

For the specific values z=0 and z=1, Maxima simplifies:

          (%i4) assume(a>0,b>0)$
          (%i5) beta_incomplete_regularized(a,b,0);
          (%o5)                                 0
          (%i6) beta_incomplete_regularized(a,b,1);
          (%o6)                                 1

Numerical evaluation for real and complex arguments in float and bigfloat precision:

          (%i7) beta_incomplete_regularized(0.12,0.43,0.9);
          (%o7)                         .9114011367359802
          (%i8) fpprec:32$
          (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0);
          (%o9)               9.1140113673598075519946998779975b-1
          (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
          (%o10)             .2865367499935403 %i - 0.122995963334684
          (%i11) fpprec:20$
          (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
          (%o12)      2.8653674999354036142b-1 %i - 1.2299596333468400163b-1

Expansion, when beta_expand is true:

          (%i13) beta_incomplete_regularized(a+1,b,z);
                                                                      b  a
                                                               (1 - z)  z
          (%o13)        beta_incomplete_regularized(a, b, z) - ------------
                                                               a beta(a, b)
          (%i14) beta_incomplete_regularized(a-1,b,z);
                                                                    b  a - 1
                                                             (1 - z)  z
          (%o14)   beta_incomplete_regularized(a, b, z) - ----------------------
                                                          beta(a, b) (b + a - 1)

The derivative and the integral wrt <z>:

          (%i15) diff(beta_incomplete_regularized(a,b,z),z);
                                               b - 1  a - 1
                                        (1 - z)      z
          (%o15)                        -------------------
                                            beta(a, b)

          (%i16) integrate(beta_incomplete_regularized(a,b,z),z);
          (%o16) beta_incomplete_regularized(a, b, z) z
                                     a beta_incomplete_regularized(a + 1, b, z)
                                   - ------------------------------------------
                                                       b + a

(%o1)                                true
(%i2) 

Beta_incomplete_regularized Example

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