Beta - Maple Help

Beta

Beta function

Calling Sequence

 Beta(x, y) $\mathrm{Β}\left(x,y\right)$

Parameters

 x - algebraic expression y - algebraic expression

Description

 • The Beta(x,y) function (Beta function) is defined in general as follows:

$\mathrm{Β}\left(x,y\right)=\frac{\mathrm{\Gamma }\left(x\right)\mathrm{\Gamma }\left(y\right)}{\mathrm{\Gamma }\left(x+y\right)}$

with the following exceptions due to GAMMA being singluar at non-positive integers:

 • When x+y is a non-positive integer but x and y are not, then Beta(x,y) is 0.
 • If x is a non-positive integer then Beta(x,y) is defined by the limit:

$\mathrm{Β}\left(x,y\right)=\underset{t\to 0}{lim}\frac{\mathrm{\Gamma }\left(x+t\right)\mathrm{\Gamma }\left(y\right)}{\mathrm{\Gamma }\left(x+t+y\right)}$

 • If y is a non-positive integer but x is not, then Beta(x,y) is defined by the symmetry relation Beta(x,y) = Beta(y,x), and the above limit is used.
 • In the cases above where the limit is computed and is finite - for example, when x and x+y are non-positive integers but y>0 - Maple signals the invalid_operation numeric event, allowing the user to control this singular behavior by catching the event. For more information, see numeric_events.
 • Note that Beta(x,y) can be represented by the following integral:

$\mathrm{Β}\left(p,q\right)={\int }_{0}^{1}{x}^{p-1}{\left(1-x\right)}^{q-1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx$

when Re(p) > 0 and Re(q) > 0.

 • Also Beta(x,y) is related to the binomial coefficient via Beta(x,y) * binomial(x+y, x) = (x+y)/x/y.
 • You can enter the command Beta using either the 1-D or 2-D calling sequence. For example, Beta(1, 2) is equivalent to $\mathrm{Β}\left(1,2\right)$.

Examples

 > $\mathrm{Β}\left(1,2\right)$
 $\frac{{1}}{{2}}$ (1)
 > $\mathrm{Β}\left(1.2+3.4I,-2.1+5.7I\right)$
 ${0.6600944470}{-}{1.126821143}{}{I}$ (2)
 > $\mathrm{Β}\left(-\frac{3}{2},-\frac{5}{2}\right)$
 ${0}$ (3)
 > $\mathrm{NumericStatus}\left(\mathrm{invalid_operation}=\mathrm{false}\right):$
 > $\mathrm{Β}\left(-3,2\right)$
 $\frac{{1}}{{6}}$ (4)
 > $\mathrm{NumericStatus}\left(\mathrm{invalid_operation}\right)$
 ${\mathrm{true}}$ (5)