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_generalized Calculator

Beta_incomplete_generalized

Function: beta_incomplete_generalized (<a>, <b>, <z1>, <z2>) The basic definition of the generalized incomplete beta function is

z2 / [ b - 1 a - 1 I (1 - t) t dt ] / z1

Maxima simplifies beta_incomplete_regularized for <a> and <b> a positive integer.

For realpart(a)>0 and z1=0 or z2=0, Maxima simplifies beta_incomplete_generalized to beta_incomplete. For realpart(b)>0 and z1=1 or <z2=1>, Maxima simplifies to an expression with beta and beta_incomplete.

Maxima evaluates beta_incomplete_regularized for real and complex values in float and bigfloat precision.

When beta_expand is true, Maxima expands beta_incomplete_generalized for a+n and a-n, <n> a positive integer.

Maxima knows the derivative of beta_incomplete_generalized with respect to the variables <a>, <b>, <z1>, and <z2> and the integrals with respect to the variables <z1> and <z2>.

Examples:

Maxima simplifies beta_incomplete_generalized for <a> and <b> a positive integer:

          (%i1) beta_incomplete_generalized(2,b,z1,z2);
                                    b                      b
                            (1 - z1)  (b z1 + 1) - (1 - z2)  (b z2 + 1)
          (%o1)             -------------------------------------------
                                             b (b + 1)

          (%i2) beta_incomplete_generalized(a,2,z1,z2);
                                               a                      a
                            (a (1 - z2) + 1) z2  - (a (1 - z1) + 1) z1
          (%o2)             -------------------------------------------
                                             a (a + 1)

          (%i3) beta_incomplete_generalized(3,2,z1,z2);
                            2      2                       2      2
                    (1 - z1)  (3 z1  + 2 z1 + 1) - (1 - z2)  (3 z2  + 2 z2 + 1)
          (%o3)     -----------------------------------------------------------
                                                12

Simplification for specific values z1=0, z2=0, z1=1, or z2=1:

          (%i4) assume(a > 0, b > 0)$
          (%i5) beta_incomplete_generalized(a,b,z1,0);
          (%o5)                    - beta_incomplete(a, b, z1)

          (%i6) beta_incomplete_generalized(a,b,0,z2);
          (%o6)                    - beta_incomplete(a, b, z2)

          (%i7) beta_incomplete_generalized(a,b,z1,1);
          (%o7)              beta(a, b) - beta_incomplete(a, b, z1)

          (%i8) beta_incomplete_generalized(a,b,1,z2);
          (%o8)              beta_incomplete(a, b, z2) - beta(a, b)

Numerical evaluation for real arguments in float or bigfloat precision:

          (%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
          (%o9)                        .09638178086368676

          (%i10) fpprec:32$
          (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
          (%o10)               9.6381780863686935309170054689964b-2

Numerical evaluation for complex arguments in float or bigfloat precision:

          (%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
          (%o11)           - .09625463003205376 %i - .003323847735353769
          (%i12) fpprec:20$
          (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
          (%o13)     - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3

Expansion for a+n or a-n, <n> a positive integer, when beta_expand is true:

          (%i14) beta_expand:true$
          (%i15) beta_incomplete_generalized(a+1,b,z1,z2);
                         b   a           b   a
                 (1 - z1)  z1  - (1 - z2)  z2
          (%o15) -----------------------------
                             b + a
                                      a beta_incomplete_generalized(a, b, z1, z2)
                                    + -------------------------------------------
                                                         b + a

          (%i16) beta_incomplete_generalized(a-1,b,z1,z2);
                 beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
          (%o16) -------------------------------------------------------
                                          1 - a
                                                    b   a - 1           b   a - 1
                                            (1 - z2)  z2      - (1 - z1)  z1
                                          - -------------------------------------
                                                            1 - a

Derivative wrt the variable <z1> and integrals wrt <z1> and <z2>:

          (%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
                                                b - 1   a - 1
          (%o17)                      - (1 - z1)      z1

          (%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
          (%o18) beta_incomplete_generalized(a, b, z1, z2) z1
                                                  + beta_incomplete(a + 1, b, z1)

          (%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
          (%o19) beta_incomplete_generalized(a, b, z1, z2) z2
                                                  - beta_incomplete(a + 1, b, z2)

(%o1)                                true
(%i2) 

Beta_incomplete_generalized Example

Related Examples

beta_incomplete_generalized

? beta_incomplete_gen...

Calculate

beta_incomplete_generalized

beta_incomplete_gener...

Calculate

beta_incomplete_generalized

? beta_incomplete_gen...

Calculate

beta_incomplete_generalized

beta_incomplete_gener...

Calculate