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_incomplete Calculator

Beta_incomplete

Function: beta_incomplete (<a>, <b>, <z>) The basic definition of the incomplete beta function (A&S 6.6.1) is

integrate(beta_incomplete(x,a,b),x,0,1);

z / [ b - 1 a - 1 I (1 - t) t dt ] / 0

This definition is possible for realpart(a)>0 and realpart(b)>0 and abs(z)<1. For other values the incomplete beta function can be defined through a generalized hypergeometric function:

gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z

(See functions.wolfram.com for a complete definition of the incomplete beta function.)

For negative integers a = -n and positive integers b=m with m<=n the incomplete beta function is defined through

m - 1 k ==== (1 - m) z n - 1 \ k z > ----------- / k! (n - k) ==== k = 0

Maxima uses this definition to simplify beta_incomplete for <a> a negative integer.

For <a> a positive integer, beta_incomplete simplifies for any argument <b> and <z> and for <b> a positive integer for any argument <a> and <z>, with the exception of <a> a negative integer.

For z=0 and realpart(a)>0, beta_incomplete has the specific value zero. For <z=1> and realpart(b)>0, beta_incomplete simplifies to the beta function beta(a,b).

Maxima evaluates beta_incomplete numerically for real and complex values in float or bigfloat precision. For the numerical evaluation an expansion of the incomplete beta function in continued fractions is used.

When beta_expand is true, Maxima expands expressions like beta_incomplete(a+n,b,z) and beta_incomplete(a-n,b,z) where n is a positive integer.

Maxima knows the derivatives of beta_incomplete with respect to the variables <a>, <b> and <z> and the integral with respect to the variable <z>.

Examples:

Simplification for <a> a positive integer:

          (%i1) beta_incomplete(2,b,z);
                                                 b
                                      1 - (1 - z)  (b z + 1)
          (%o1)                       ----------------------
                                            b (b + 1)

Simplification for <b> a positive integer:

          (%i2) beta_incomplete(a,2,z);
                                                         a
                                        (a (1 - z) + 1) z
          (%o2)                         ------------------
                                            a (a + 1)

Simplification for <a> and <b> a positive integer:

          (%i3) beta_incomplete(3,2,z);
                                                         3
                                        (3 (1 - z) + 1) z
          (%o3)                         ------------------
                                                12

<a> is a negative integer and b<=(-a), Maxima simplifies:

          (%i4) beta_incomplete(-3,1,z);
                                                 1
          (%o4)                              - ----
                                                  3
                                               3 z

For the specific values z=0 and z=1, Maxima simplifies:

          (%i5) assume(a>0,b>0)$
          (%i6) beta_incomplete(a,b,0);
          (%o6)                                 0
          (%i7) beta_incomplete(a,b,1);
          (%o7)                            beta(a, b)

Numerical evaluation in float or bigfloat precision:

          (%i8) beta_incomplete(0.25,0.50,0.9);
          (%o8)                          4.594959440269333
          (%i9)  fpprec:25$
          (%i10) beta_incomplete(0.25,0.50,0.9b0);
          (%o10)                    4.594959440269324086971203b0

For abs(z)>1 beta_incomplete returns a complex result:

          (%i11) beta_incomplete(0.25,0.50,1.7);
          (%o11)              5.244115108584249 - 1.45518047787844 %i

Results for more general complex arguments:

          (%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
          (%o14)             2.726960675662536 - .3831175704269199 %i
          (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i);
          (%o15)             13.04649635168716 %i - 5.802067956270001
          (%i16)

Expansion, when beta_expand is true:

          (%i23) beta_incomplete(a+1,b,z),beta_expand:true;
                                                                 b  a
                             a beta_incomplete(a, b, z)   (1 - z)  z
          (%o23)             -------------------------- - -----------
                                       b + a                 b + a

          (%i24) beta_incomplete(a-1,b,z),beta_expand:true;
                                                                     b  a - 1
                     beta_incomplete(a, b, z) (- b - a + 1)   (1 - z)  z
          (%o24)     -------------------------------------- - ---------------
                                     1 - a                         1 - a

Derivative and integral for beta_incomplete:

          (%i34) diff(beta_incomplete(a,b,z),z);
                                               b - 1  a - 1
          (%o34)                        (1 - z)      z
          (%i35) integrate(beta_incomplete(a,b,z),z);
          (%o35)     beta_incomplete(a, b, z) z - beta_incomplete(a + 1, b, z)
          (%i36) diff(%,z);
          (%o36)                     beta_incomplete(a, b, z)

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

(%o1)                                true
(%i2) 

Beta_incomplete Example

Related Examples

beta_incomplete-integrate

integrate(beta_incomp...

Calculate

beta_incomplete-integrate

integrate(beta_incomp...

Calculate

beta_incomplete

? beta_incomplete;

Calculate

beta_incomplete-integrate

integrate(beta_incomp...

Calculate

beta_incomplete-integrate

integrate(beta_incomp...

Calculate

beta_incomplete

? beta_incomplete;

Calculate