Sponsored links: Algebra eBooks
 

Help Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

The Maxima on-line user's manual

Algebra Calculator

Search:

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).

r: 100;
beta(theta) := min(12/25*theta,%pi/4);
plot2d(beta(theta),[theta,0,2*%pi]);
d1(theta) := r/sin(beta(theta))*(2*cos(beta(theta))-cos(theta)-1);
d1p(theta) := d1(theta) + r*tan(beta(theta)/2);
x(theta) := r*sin(theta+beta(theta))+d1p(theta)*cos(theta+beta(theta));
y(theta) := r*(1-cos(theta+beta(theta)))+d1p(theta)*sin(theta+beta(theta));
dist(theta) := sqrt(x(theta)^2+y(theta)^2);
plot2d(dist(theta),[theta,0,2*%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-kill-transpose

kill (eq1,eq2,eq3,eq4...

v1:[1,2,0,1];

n:transpose(v1);

Calculate

beta-collectterms-cos-sin-solve

x:a*cos(alpha) + x0;

y:b*sin(alpha) + y0;

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

Calculate

beta-coefmatrix-lambda

eq1:x=lambda;

+alpha;

+beta;

Calculate

beta-define-integrate

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

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

define(g(x), integrat...

Calculate

beta-define-integrate

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

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

define(g(x), integrat...

Calculate

beta-expand-float-solve

R1:820;

R2:820;

vth:0.6;

Calculate

beta-gamma-numer-sin

alpha:58;

beta:75;

gamma:180-(alpha+beta);

Calculate

beta-coefmatrix-echelon-rank-transpose

eq1:x=lambda+alfa+beta;

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

eq3:z=lambda+2*alfa;

Calculate

beta-coefmatrix-rank-transpose

eq1:x=lambda+alpha+beta;

eq2:y=lambda-alpha+3*...

eq3:z=lambda+2*alpha;

Calculate

beta-cos-matrix-sin-tan

n1 = matrix([cos(alph...

n2 = matrix([cos(beta...

e1 = matrix([1], [1/t...

Calculate