### The Maxima on-line user's manual

Algebra Calculator

#### Beta_incomplete_generalized

Function: beta_incomplete_generalized (<a>, <b>, <z1>, <z2>) The basic definition of the generalized incomplete beta function is

z2 / [ b - 1 a - 1 I (1 - t) t dt ] / z1

Maxima simplifies `beta_incomplete_regularized` for <a> and <b> a positive integer.

For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies `beta_incomplete_generalized` to `beta_incomplete`. For realpart(b)>0 and z1=1 or <z2=1>, Maxima simplifies to an expression with `beta` and `beta_incomplete`.

Maxima evaluates `beta_incomplete_regularized` for real and complex values in float and bigfloat precision.

When `beta_expand` is `true`, Maxima expands `beta_incomplete_generalized` for a+n and a-n, <n> a positive integer.

Maxima knows the derivative of `beta_incomplete_generalized` with respect to the variables <a>, <b>, <z1>, and <z2> and the integrals with respect to the variables <z1> and <z2>.

Examples:

Maxima simplifies `beta_incomplete_generalized` for <a> and <b> a positive integer:

```          (%i1) beta_incomplete_generalized(2,b,z1,z2);
b                      b
(1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
(%o1)             -------------------------------------------
b (b + 1)```

```          (%i2) beta_incomplete_generalized(a,2,z1,z2);
a                      a
(a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
(%o2)             -------------------------------------------
a (a + 1)```

```          (%i3) beta_incomplete_generalized(3,2,z1,z2);
2      2                       2      2
(1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
(%o3)     -----------------------------------------------------------
12```

Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

```          (%i4) assume(a > 0, b > 0)\$
(%i5) beta_incomplete_generalized(a,b,z1,0);
(%o5)                    - beta_incomplete(a, b, z1)```

```          (%i6) beta_incomplete_generalized(a,b,0,z2);
(%o6)                    - beta_incomplete(a, b, z2)```

```          (%i7) beta_incomplete_generalized(a,b,z1,1);
(%o7)              beta(a, b) - beta_incomplete(a, b, z1)```

```          (%i8) beta_incomplete_generalized(a,b,1,z2);
(%o8)              beta_incomplete(a, b, z2) - beta(a, b)```

Numerical evaluation for real arguments in float or bigfloat precision:

```          (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
(%o9)                        .09638178086368676```

```          (%i10) fpprec:32\$
(%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
(%o10)               9.6381780863686935309170054689964b-2```

Numerical evaluation for complex arguments in float or bigfloat precision:

```          (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
(%o11)           - .09625463003205376 %i - .003323847735353769
(%i12) fpprec:20\$
(%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
(%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3```

Expansion for a+n or a-n, <n> a positive integer, when `beta_expand` is `true`:

```          (%i14) beta_expand:true\$
(%i15) beta_incomplete_generalized(a+1,b,z1,z2);
b   a           b   a
(1 - z1)  z1  - (1 - z2)  z2
(%o15) -----------------------------
b + a
a beta_incomplete_generalized(a, b, z1, z2)
+ -------------------------------------------
b + a```

```          (%i16) beta_incomplete_generalized(a-1,b,z1,z2);
beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
(%o16) -------------------------------------------------------
1 - a
b   a - 1           b   a - 1
(1 - z2)  z2      - (1 - z1)  z1
- -------------------------------------
1 - a```

Derivative wrt the variable <z1> and integrals wrt <z1> and <z2>:

```          (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
b - 1   a - 1
(%o17)                      - (1 - z1)      z1```

```          (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
(%o18) beta_incomplete_generalized(a, b, z1, z2) z1
+ beta_incomplete(a + 1, b, z1)```

```          (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
(%o19) beta_incomplete_generalized(a, b, z1, z2) z2
- beta_incomplete(a + 1, b, z2)```

```(%o1)                                true
(%i2) ```

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