Dilogarithm

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the dilogarithm (or Spence's function), denoted as Li2(z), is a particular case of the <a href="/facts/Polylogarithm/CKrkJMKn">polylogarithm</a>. Two related <a href="/facts/Special_functions/JETSeucA">special functions</a> are referred to as Spence's function, the dilogarithm itself:

Li
          
            2
          
        
        ⁡
        (
        z
        )
        =
        −
        
          ∫
          
            0
          
          
            z
          
        
        
          
            
              ln
              ⁡
              (
              1
              −
              u
              )
            
            u
          
        
        
        d
        u
        
          , 
        
        z
        ∈
        
          C
        
      
    
    {\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C} }

and its reflection.
For |z| ≤ 1, an <a href="/facts/Infinite_series/yDnlsgIe">infinite series</a> also applies (the integral definition constitutes its analytical extension to the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>):

Li
          
            2
          
        
        ⁡
        (
        z
        )
        =
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              z
              
                k
              
            
            
              k
              
                2
              
            
          
        
        .
      
    
    {\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.}

Alternatively, the dilogarithm function is sometimes defined as

∫
          
            1
          
          
            v
          
        
        
          
            
              ln
              ⁡
              t
            
            
              1
              −
              t
            
          
        
        d
        t
        =
        
          Li
          
            2
          
        
        ⁡
        (
        1
        −
        v
        )
        .
      
    
    {\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).}

In <a href="/facts/Hyperbolic_geometry/DjTMKdgy">hyperbolic geometry</a> the dilogarithm can be used to compute the volume of an ideal simplex. Specifically, a simplex whose vertices have <a href="/facts/Cross_ratio/tj4DK3Vk">cross ratio</a> z has <a href="/facts/Hyperbolic_volume/BVGBPBjK">hyperbolic volume</a>

D
        (
        z
        )
        =
        Im
        ⁡
        
          Li
          
            2
          
        
        ⁡
        (
        z
        )
        +
        arg
        ⁡
        (
        1
        −
        z
        )
        log
        ⁡
        
          |
        
        z
        
          |
        
        .
      
    
    {\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.}

The function D(z) is sometimes called the Bloch-Wigner function. <a href="/facts/Lobachevsky%2527s_function/1mDkvwcL">Lobachevsky's function</a> and <a href="/facts/Clausen%2527s_function/1mDkvwcL">Clausen's function</a> are closely related functions.
<a href="/facts/William_Spence_(mathematician)/BcGD5cRt">William Spence</a>, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with <a href="/facts/John_Galt_(novelist)/JkVSPp9A">John Galt</a>, who later wrote a biographical essay on Spence.

Dilogarithm open-in-new

Dilogarithm