Half-exponential function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a half-exponential function is a <a href="/facts/Functional_square_root/C3qCTVCw">functional square root</a> of an <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>. That is, a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
  
    
      
        f
      
    
    {\displaystyle f}
  
 such that 
  
    
      
        f
      
    
    {\displaystyle f}
  
  <a href="/facts/Function_composition/r5Pvnmth">composed</a> with itself results in an exponential function:

f
        
          
            (
          
        
        f
        (
        x
        )
        
          
            )
          
        
        =
        a
        
          b
          
            x
          
        
        ,
      
    
    {\displaystyle f{\bigl (}f(x){\bigr )}=ab^{x},}

for some constants 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and 
  
    
      
        b
      
    
    {\displaystyle b}
  
.
</p><p><a href="/facts/Hellmuth_Kneser/IhDlnYET">Hellmuth Kneser</a> first proposed a <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic</a> construction of the solution of 
  
    
      
        f
        
          
            (
          
        
        f
        (
        x
        )
        
          
            )
          
        
        =
        
          e
          
            x
          
        
      
    
    {\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}
  
 in 1950. It is closely related to the problem of extending <a href="/facts/Tetration/5rKeVVYv">tetration</a> to non-integer values; the value of

1
              2
            
          
        
        a
      
    
    {\displaystyle {}^{\frac {1}{2}}a}
  
 can be understood as the value of 
  
    
      
        f
        
          
            (
          
        
        1
        )
      
    
    {\displaystyle f{\bigl (}1)}
  
, where 
  
    
      
        f
        
          
            (
          
        
        x
        )
      
    
    {\displaystyle f{\bigl (}x)}
  
 satisfies 
  
    
      
        f
        
          
            (
          
        
        f
        (
        x
        )
        
          
            )
          
        
        =
        
          a
          
            x
          
        
      
    
    {\displaystyle f{\bigl (}f(x){\bigr )}=a^{x}}
  
. Example values from Kneser's solution of 
  
    
      
        f
        
          
            (
          
        
        f
        (
        x
        )
        
          
            )
          
        
        =
        
          e
          
            x
          
        
      
    
    {\displaystyle f{\bigl (}f(x){\bigr )}=e^{x}}
  
 include 
  
    
      
        f
        
          
            (
          
        
        0
        )
        ≈
        0.49856
      
    
    {\displaystyle f{\bigl (}0)\approx 0.49856}
  
 and 
  
    
      
        f
        
          
            (
          
        
        1
        )
        ≈
        1.64635
      
    
    {\displaystyle f{\bigl (}1)\approx 1.64635}
  
.
</p>

Half-exponential function open-in-new

Half-exponential function