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The Maxima on-line user's manual

Algebra Calculator

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Beta Calculator

Beta

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

beta(x) := 2/x * sin(x/2);
plot2d([beta(x),%pi/2,%pi/4],[x,0.1,2*%pi]);

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) 

Beta Example

Related Examples

beta-gamma-matrix-transpose

R1:matrix([(COS(alfa)...

Pa:[X0A,Y0A,Z0A];

Pa:transpose(Pa);

Calculate

beta-collectterms-cos-sin

x=a*cos(alpha) + x0;

y=b*sin(alpha) + y0;

xp=x*cos(beta) - y*si...

Calculate

beta-coefmatrix

eq1: x=lambda+alfa+beta;

eq2: y=lambda-alfa+be...

eq3: z=lambda+ alfa;

Calculate

beta-realpart

aa:(1+i*nu)*(U+i*V)-(...

realpart((1+i*nu)*(U+...

Calculate

beta-kill

kill(eq1,eq1,eq3,eq4,...

Calculate

beta-coefmatrix

eq1: x=lambda+alfa+beta;

eq2: y=lambda-alfa+be...

eq3: z=lambda+ alfa;

Calculate

beta-cos-expand-sin

x:a*cos(alpha) + x0;

y:b*sin(alpha) + y0;

xp:x*cos(beta) - y*si...

Calculate

beta-cosh-matrix-ratsimp-sinh-transpose

iT : matrix( [ cosh(R...

T : matrix( [ cosh(R)...

R : matrix( [ cos(2*b...

Calculate

beta-cos-diff-sin-sqrt

x:a*cos(t)+xc;

y:b*sin(t)+yc;

xp:x*cos(beta)-y*sin(...

Calculate

beta-cos-diff-sin

x:a*cos(t)+xc;

y:b*sin(t)+yc;

xp:x*cos(beta)-y*sin(...

Calculate