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

Algebra Calculator

#### Beta

Function: beta (<a>, <b>) The beta function is defined as `gamma(a) gamma(b)/gamma(a+b)` (A&S 6.2.1).

Maxima simplifies the beta function for positive integers and rational numbers, which sum to an integer. When `beta_args_sum_to_integer` is `true`, Maxima simplifies also general expressions which sum to an integer.

For <a> or <b> equal to zero the beta function is not defined.

In general the beta function is not defined for negative integers as an argument. The exception is for <a=-n>, <n> a positive integer and <b> a positive integer with <b<=n>, it is possible to define an analytic continuation. Maxima gives for this case a result.

When `beta_expand` is `true`, expressions like `beta(a+n,b)` and `beta(a-n,b)` or `beta(a,b+n)` and `beta(a,b-n)` with `n` an integer are simplified.

Maxima can evaluate the beta function for real and complex values in float and bigfloat precision. For numerical evaluation Maxima uses `log_gamma`:

- log_gamma(b + a) + log_gamma(b) + log_gamma(a) %e

Maxima knows that the beta function is symmetric and has mirror symmetry.

Maxima knows the derivatives of the beta function with respect to <a> or <b>.

To express the beta function as a ratio of gamma functions see `makegamma`.

Examples:

Simplification, when one of the arguments is an integer:

```          (%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
1   9      1
(%o1)                         [--, -, ---------]
12  4  a (a + 1)```

Simplification for two rational numbers as arguments which sum to an integer:

```          (%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
3 %pi   2 %pi
(%o2)                    [-----, -------, sqrt(2) %pi]
8    sqrt(3)```

When setting `beta_args_sum_to_integer` to `true` more general expression are simplified, when the sum of the arguments is an integer:

```          (%i3) beta_args_sum_to_integer:true\$
(%i4) beta(a+1,-a+2);
%pi (a - 1) a
(%o4)                         ------------------
2 sin(%pi (2 - a))```

The possible results, when one of the arguments is a negative integer:

```          (%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
1  1    1
(%o5)                            [- -, -, - -]
3  6    3```

`beta(a+n,b)` or `beta(a-n)` with `n` an integer simplifies when `beta_expand` is `true`:

```          (%i6) beta_expand:true\$
(%i7) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
a beta(a, b)  beta(a, b) (b + a - 1)  a
(%o7)              [------------, ----------------------, -]
b + a              a - 1           b```

Beta is not definied, when one of the arguments is zero:

```          (%i7) beta(0,b);
beta: expected nonzero arguments; found 0, b
-- an error.  To debug this try debugmode(true);```

Numercial evaluation for real and complex arguments in float or bigfloat precision:

```          (%i8) beta(2.5,2.3);
(%o8) .08694748611299981```

```          (%i9) beta(2.5,1.4+%i);
(%o9) 0.0640144950796695 - .1502078053286415 %i```

```          (%i10) beta(2.5b0,2.3b0);
(%o10) 8.694748611299969b-2```

```          (%i11) beta(2.5b0,1.4b0+%i);
(%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i```

Beta is symmetric and has mirror symmetry:

```          (%i14) beta(a,b)-beta(b,a);
(%o14)                                 0
(%i15) declare(a,complex,b,complex)\$
(%i16) conjugate(beta(a,b));
(%o16)                 beta(conjugate(a), conjugate(b))```

```     The derivative of the beta function wrt `a`:
(%i17) diff(beta(a,b),a);
(%o17)               - beta(a, b) (psi (b + a) - psi (a))
0             0```

There are also some inexact matches for `beta`. Try `?? beta` to see them.

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

### Related Examples

##### beta-cos-lambda-linsolve-sin

eq1:-b*phi*cos(beta)*...

eq2:b*phi*cos(beta)*c...

linsolve([eq1,eq2],[x]);

Calculate

##### beta-cos-gamma-ratsimp-sin-sqrt

ratsimp((sqrt((cos(al...

Calculate

##### beta-define-integrate

define(p(n, k, x), x^...

define(f(x, y), integ...

define(g(x), integrat...

Calculate

##### beta-ev-false-floor-numer-ratprint-solve

"*"/* Auf einer Anhöh...

"*"/* Eingabe der Win...

alpha:14;

Calculate

##### beta-coefmatrix-echelon-rank-transpose

eq1:x=lambda+alfa+beta;

eq2:y=lambda-alfa+3*b...

eq3:z=lambda+2*alfa;

Calculate

##### beta-cos-matrix

a:matrix([1,0,0],[0,c...

b:matrix([cos(beta),0...

rotate=a.b;

Calculate

x0:%i;

alpha:0;

H1p(x):=-9/200;

Calculate

x0:%i;

alpha:0;

H1p(x):=-9/200;

Calculate

##### beta-cos-sin-solve

solve([L1*sin((%pi/2)...

Calculate

##### beta-cos-min-plot2d-sin-sqrt-tan

r: 100;

beta(theta) := min(%p...

d1(theta) := r/sin(be...

Calculate