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

Algebra 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) ```

### Related Examples

##### beta_incomplete_regularized-define-diff

define(f(a, b, z), di...

f(3.2, 4.5, 0.7);

hypergeometric_regula...

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a:1;

b:19;

p:a/(a+b);

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a:400;

b:100;

p:a/(a+b);

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a:1;

b:500;

r:a/(a+b)+0.05;

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a:90;

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r:a/(a+b)+0.05;

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##### beta_incomplete_regularized

beta_incomplete_regul...

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a:20;

b:20;

r:a/(a+b)+0.05;

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a:1000;

b:100;

p:a/(a+b);

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a:2;

b:.1;

p:a/(a+b);

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a:3;

b:1;

p:a/(a+b);

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