In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.
 When 
  
    
      
        R
        e
        (
        α
        )
        >
        0
        ,
        R
        e
        (
        β
        )
        >
        0
        ,
        R
        e
        (
        γ
        )
        >
        −
        min
        
          (
          
            
              
                1
                n
              
            
            ,
            
              
                
                  R
                  e
                  (
                  α
                  )
                
                
                  n
                  −
                  1
                
              
            
            ,
            
              
                
                  R
                  e
                  (
                  β
                  )
                
                
                  n
                  −
                  1
                
              
            
          
          )
        
      
    
    {\displaystyle Re(\alpha )>0,Re(\beta )>0,Re(\gamma )>-\min \left({\frac {1}{n}},{\frac {Re(\alpha )}{n-1}},{\frac {Re(\beta )}{n-1}}\right)}
  
, we have
 Selberg's formula  implies Dixon's identity for well poised hypergeometric series, and some special cases of Dyson's conjecture. This is a corollary of Aomoto.
 Aomoto proved a slightly more general integral formula. With the same conditions as Selberg's formula,
 A proof is found in Chapter 8 of Andrews, Askey & Roy (1999).
 When 
  
    
      
        R
        e
        (
        γ
        )
        >
        −
        1
        
          /
        
        n
      
    
    {\displaystyle Re(\gamma )>-1/n}
  
, 
 It is a corollary of Selberg, by setting 
  
    
      
        α
        =
        β
      
    
    {\displaystyle \alpha =\beta }
  
, and change of variables with 
  
    
      
        
          t
          
            i
          
        
        =
        
          
            
              1
              +
              
                t
                
                  i
                
                ′
              
              
                /
              
              
                
                  2
                  α
                
              
            
            2
          
        
      
    
    {\displaystyle t_{i}={\frac {1+t'_{i}/{\sqrt {2\alpha }}}{2}}}
  
, then taking 
  
    
      
        α
        →
        ∞
      
    
    {\displaystyle \alpha \to \infty }
  
. 
 This was conjectured by Mehta & Dyson (1963), who were unaware of Selberg's earlier work.
 It is the partition function for a gas of point charges moving on a line that are attracted to the origin.
 Macdonald (1982) conjectured the following extension of Mehta's integral to all finite root systems, Mehta's original case corresponding to the An−1 root system.
 The product is over the  roots r of the roots system and the numbers dj are the degrees of the generators of the ring of invariants of the reflection group. 
Opdam (1989) gave a uniform proof for all crystallographic reflection groups. Several years later he proved it in full generality, making use of computer-aided calculations by Garvan.