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Beta
Beta[
a
,
b
] gives the Euler beta function  .
Beta[
z
,
a
,
b
] gives the incomplete beta function  .
Mathematical function (see Section A.3.10).
 .
 .
Beta[
z
,
a
,
b
] has a branch cut discontinuity in the complex z plane running from  to  .
Beta[
,
,
a
,
b
] gives the generalized incomplete beta function  .
Note that the arguments in the incomplete form of Beta are arranged differently from those in the incomplete form of Gamma.
In TraditionalForm, Beta is output using \[CapitalBeta].
See the Mathematica book: Section 3.2.10.
See also: InverseBetaRegularized.
Further Examples
Here is a symbolic result.
In[1]:= 
Out[1]= 
This is the derivative.
In[2]:= 
Out[2]= 
This is the second derivative.
In[3]:= 
Out[3]= 
For a and b positive real, this just gives an alternate definition of the Beta function.
In[4]:= 
Out[4]= 
In[5]:= 
Out[5]= 
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