J. Boersma

J. P. Dempsey

English

Legendre’s chi-function, 

  
    
      
        
          χ
          n
        
      
      (
      z
      )
      =
      
        Σ
        
          k
          =
          0
        
        ∞
      
      
        
          z
          
            2
            k
            +
            1
          
        
      
      
        /
      
      
        (
        2
        k
        +
        1
        
          )
          n
        
      
    
    {\chi _n}(z) = \Sigma _{k = 0}^\infty {z^{2k + 1}}/{(2k + 1)^n}
  

, is reexpanded in a power series in powers of 

  
    
      log
      ⁡
      z
    
    \log z
  

. The expansion obtained is well suited for the computation of 

  
    
      
        
          χ
          n
        
      
      (
      z
      )
    
    {\chi _n}(z)
  

 in the two cases of real z close to 1, and 

  
    
      z
      =
      
        
          e
          
            i
            α
          
        
      
      ,
      α
      ∈
      
        R
      
    
    z = {e^{i\alpha }},\alpha \in \mathbb {R}
  

. For 

  
    
      n
      =
      2
    
    n = 2
  

 and 

  
    
      n
      =
      3
    
    n = 3
  

, the present computational procedure is shown to be superior to the procedure recently proposed by Dempsey, Liu, and Dempsey, which uses Plana’s summation formula.</p

Crossref

Mathematics of Computation

On the numerical evaluation of Legendre’s chi-function

https://pure.tue.nl/ws/files/1670806/353771.pdf

On the numerical evaluation of Legendre’s chi-function

Abstract

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Crossref