Polylogarithm

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a <a href="/facts/Special_function/JETSeucA">special function</a> Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an <a href="/facts/Elementary_function/wVFyDwu8">elementary function</a> such as the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> or a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a>. In <a href="/facts/Quantum_statistics/hJaSBe1l">quantum statistics</a>, the polylogarithm function appears as the closed form of <a href="/facts/Integral/V7lyV997">integrals</a> of the <a href="/facts/Fermi%E2%80%93Dirac_distribution/aLo4bton">Fermi–Dirac distribution</a> and the <a href="/facts/Bose%E2%80%93Einstein_distribution/Utb2jN3n">Bose–Einstein distribution</a>, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In <a href="/facts/Quantum_electrodynamics/zPteAr6G">quantum electrodynamics</a>, polylogarithms of positive <a href="/facts/Integer/PUwwrawx">integer</a> order arise in the calculation of processes represented by higher-order <a href="/facts/Feynman_diagram/BRFNvDdr">Feynman diagrams</a>.
The polylogarithm function is equivalent to the <a href="/facts/Hurwitz_zeta_function/4EH7sKXB">Hurwitz zeta function</a> — either <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> can be expressed in terms of the other — and both functions are special cases of the <a href="/facts/Lerch_transcendent/26RvsFok">Lerch transcendent</a>. Polylogarithms should not be confused with <a href="/facts/Polylogarithmic_function/6zrJ4Ger">polylogarithmic functions</a>, nor with the <a href="/facts/Offset_logarithmic_integral/P3xwqRlA">offset logarithmic integral</a> Li(z), which has the same notation without the subscript.

The polylogarithm function is defined by a <a href="/facts/Power_series/vYh6sTps">power series</a> in z, which is also a <a href="/facts/Dirichlet_series/uIAUaLcd">Dirichlet series</a> in s:

Li
          
            s
          
        
        ⁡
        (
        z
        )
        =
        
          ∑
          
            k
            =
            1
          
          
            ∞
          
        
        
          
            
              z
              
                k
              
            
            
              k
              
                s
              
            
          
        
        =
        z
        +
        
          
            
              z
              
                2
              
            
            
              2
              
                s
              
            
          
        
        +
        
          
            
              z
              
                3
              
            
            
              3
              
                s
              
            
          
        
        +
        ⋯
      
    
    {\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}=z+{z^{2} \over 2^{s}}+{z^{3} \over 3^{s}}+\cdots }

This definition is valid for arbitrary <a href="/facts/Complex_numbers/2w7mqI7e">complex</a> order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a>. (Here the denominator ks is understood as exp(s ln k)). The special case s = 1 involves the ordinary <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a>, Li1(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the <a href="/facts/Dilogarithm/lsnCj3Gb">dilogarithm</a> (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated <a href="/facts/Indefinite_integral/5qDsudIv">integral</a> of itself:

Li
          
            s
            +
            1
          
        
        ⁡
        (
        z
        )
        =
        
          ∫
          
            0
          
          
            z
          
        
        
          
            
              
                Li
                
                  s
                
              
              ⁡
              (
              t
              )
            
            t
          
        
        d
        t
      
    
    {\displaystyle \operatorname {Li} _{s+1}(z)=\int _{0}^{z}{\frac {\operatorname {Li} _{s}(t)}{t}}dt}

thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a <a href="/facts/Rational_function/2nJGu0Ko">rational function</a>.

Polylogarithm open-in-new

Polylogarithm