Buchstab function

The Buchstab function (or Buchstab's function) is the unique continuous function 
 
 
 
 ω
 :
 
 
 R
 
 
 ≥
 1
 
 
 →
 
 
 R
 
 
 >
 0
 
 
 
 
 {\displaystyle \omega :\mathbb {R} _{\geq 1}\rightarrow \mathbb {R} _{>0}}
 
 defined by the <a href="/facts/Delay_differential_equation/fM5wzXKr">delay differential equation</a>

ω
        (
        u
        )
        =
        
          
            1
            u
          
        
        ,
        
        
        
        1
        ≤
        u
        ≤
        2
        ,
      
    
    {\displaystyle \omega (u)={\frac {1}{u}},\qquad \qquad \qquad 1\leq u\leq 2,}

d
            
              d
              u
            
          
        
        (
        u
        ω
        (
        u
        )
        )
        =
        ω
        (
        u
        −
        1
        )
        ,
        
        u
        ≥
        2.
      
    
    {\displaystyle {\frac {d}{du}}(u\omega (u))=\omega (u-1),\qquad u\geq 2.}

In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. It is named after <a href="/facts/Alexander_Buchstab/tE0H0PQR">Alexander Buchstab</a>, who wrote about it in 1937.

Buchstab function open-in-new

Buchstab function