Beta function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the beta function, also called the <a href="/facts/Euler_integral_(disambiguation)/8L8aNjNF">Euler integral</a> of the first kind, is a <a href="/facts/Special_function/JETSeucA">special function</a> that is closely related to the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> and to <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficients</a>. It is defined by the <a href="/facts/Integral/V7lyV997">integral</a>

B
        
        (
        
          z
          
            1
          
        
        ,
        
          z
          
            2
          
        
        )
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          t
          
            
              z
              
                1
              
            
            −
            1
          
        
        (
        1
        −
        t
        
          )
          
            
              z
              
                2
              
            
            −
            1
          
        
        
        d
        t
      
    
    {\displaystyle \mathrm {B} (z_{1},z_{2})=\int _{0}^{1}t^{z_{1}-1}(1-t)^{z_{2}-1}\,dt}

for <a href="/facts/Complex_number/2w7mqI7e">complex number</a> inputs

z
 
 1
 
 
 ,
 
 z
 
 2
 
 
 
 
 {\displaystyle z_{1},z_{2}}
 
 such that 
 
 
 
 Re
 ⁡
 (
 
 z
 
 1
 
 
 )
 ,
 Re
 ⁡
 (
 
 z
 
 2
 
 
 )
 >
 0
 
 
 {\displaystyle \operatorname {Re} (z_{1}),\operatorname {Re} (z_{2})>0}
 
.
The beta function was studied by <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> and <a href="/facts/Adrien-Marie_Legendre/2a7CAolM">Adrien-Marie Legendre</a> and was given its name by <a href="/facts/Jacques_Philippe_Marie_Binet/7xEjWJql">Jacques Binet</a>; its symbol Β is a <a href="/facts/Greek_alphabet/1cHwB8Dg">Greek</a> capital <a href="/facts/Beta_(letter)/oTWMRsHj">beta</a>.

Beta function open-in-new

Beta function