Lerch transcendent

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Lerch transcendent, is a <a href="/facts/Special_function/JETSeucA">special function</a> that generalizes the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a> and the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a>.  It is named after Czech mathematician <a href="/facts/Mathias_Lerch/IaSB4XfL">Mathias Lerch</a>, who published a paper about a similar function in 1887. The Lerch transcendent, is given by:
</p>

Φ
        (
        z
        ,
        s
        ,
        α
        )
        =
        
          ∑
          
            n
            =
            0
          
          
            ∞
          
        
        
          
            
              z
              
                n
              
            
            
              (
              n
              +
              α
              
                )
                
                  s
                
              
            
          
        
      
    
    {\displaystyle \Phi (z,s,\alpha )=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+\alpha )^{s}}}}
  
.
<p>It only converges for any real number 
  
    
      
        α
        >
        0
      
    
    {\displaystyle \alpha >0}
  
, where 
  
    
      
        
          |
        
        z
        
          |
        
        <
        1
      
    
    {\displaystyle |z|<1}
  
, or 
  
    
      
        
          
            R
          
        
        (
        s
        )
        >
        1
      
    
    {\displaystyle {\mathfrak {R}}(s)>1}
  
, and 
  
    
      
        
          |
        
        z
        
          |
        
        =
        1
      
    
    {\displaystyle |z|=1}
  
.
</p>

Lerch transcendent open-in-new

Lerch transcendent