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As an exercise ofin writing a good Compile function, I want to do the simple task of coding a routine that outputs the real part of the dilogarithm function reLi2[z], given a complex number z as input. I would like some feedback in my code below that does the job.

I am following the strategy outlined in Celestial Mech. Dynam. Astron. 62 (1): 93–98. TheThe idea is to divide the complex $z$-plane into four regions, as I show in the figure below, and to implement a different approximation in each region.

  1. In region 1, approximate the dilogarithm by the defining sum: $\text{Li}_2(z) \approx \sum_{k=1}^\infty x^k/k^2$$\operatorname{Li}_2(z) \approx \sum_{k=1}^\infty x^k/k^2$. Here is the code:

    realRegion1 = Compile[{{x, _Real}}, Sum[x^k/k^2, {k, 1., 23}]];
    
  2. In region 2, approximate the dilogarithm by evaluating the integral $\text{Li}_2(z) \approx -\int_0^1 \frac{\ln(1-z t)}{t}\,dt$$\operatorname{Li}_2(z) \approx -\int_0^1 \frac{\ln(1-z t)}{t}\,dt$ by Gaussian quadrature (9 divisions):

    With[{div = 9},
      With[{x = Sort[formalX /. Solve[LegendreP[div, formalX] == 0, formalX] // N]},
        With[{
          y = Chop[x/2 + 1/2], 
          w = Table[2/((1 - x[[i]]^2)*Derivative[0, 1][LegendreP][div, x[[i]]]^2), 
                {i, 1, div}]},
          With[{
            realRegion2expr =
              -1/4.*Sum[
                      Chop[w[[i]]] Log[1 - 2 y[[i]] var + y[[i]]^2 var^2]/y[[i]], 
                      {i, 1, div}]},
            realRegion2 = Compile[{{var, _Real}}, realRegion2expr]]
         ]
       ]
     ];
    
  3. In region 3, apply the dilogarithm identity $\text{Li}_2(z) = -\underbrace{\text{Li}_2(1-z)}_\text{region I} - \ln(z)\ln(1-z)+\pi^2/6$$\operatorname{Li}_2(z) = -\underbrace{\operatorname{Li}_2(1-z)}_\text{region I} - \ln(z)\ln(1-z)+\pi^2/6$, where the dilogarithm in the RHS is to be evaluated in region I.

    realRegion3 = Compile[{{x, _Real}}, 
     If[x == 1, Pi^2/6., -realRegion1[1 - x] - Log[x] Log[1 - x] + Pi^2/6.],
       CompilationOptions -> {"InlineExternalDefinitions" -> True}];
    
  4. In region 4, apply the dilogarithm identity $\text{Li}_2(z) = -\underbrace{\text{Li}_2(1/z)}_\text{region I,II,III} - \frac{1}{2}\ln^2(-z)-\pi^2/6$$\operatorname{Li}_2(z) = -\underbrace{\operatorname{Li}_2(1/z)}_\text{region I,II,III} - \frac{1}{2}\ln^2(-z)-\pi^2/6$, where the dilogarithm in the RHS is to be appropriately evaluated in region I, II or III depending on the value of $1/z$.

    So first, I need to put together the functions realRegion1, realRegion2 and realRegion3 so that it correctly evaluates on the real line segment $-1 \leq z \leq +1$ appropriately:

    realSegment = 
      Compile[{{x, _Real}},
        If[-0.5 <= x <= 0.5, realRegion1[x],
          If[x <= 0, realRegion2[x], realRegion3[x]]
        ], CompilationOptions -> {"InlineExternalDefinitions" -> True}
      ];
    

The output is quite satisfactory. You can compare Plot[reLi2[x], {x, -5, 5}] with Plot[Re@PolyLog[2Plot[Re @ PolyLog[2, x], {x, -5, 5}].

As an exercise of writing a good Compile function, I want to do the simple task of coding a routine that outputs the real part of the dilogarithm function reLi2[z], given a complex number z as input. I would like some feedback in my code below that does the job.

I am following the strategy outlined in Celestial Mech. Dynam. Astron. 62 (1): 93–98. The idea is to divide the complex $z$-plane into four regions, as I show in the figure below, and to implement a different approximation in each region.

  1. In region 1, approximate the dilogarithm by the defining sum: $\text{Li}_2(z) \approx \sum_{k=1}^\infty x^k/k^2$. Here is the code:

    realRegion1 = Compile[{{x, _Real}}, Sum[x^k/k^2, {k, 1., 23}]];
    
  2. In region 2, approximate the dilogarithm by evaluating the integral $\text{Li}_2(z) \approx -\int_0^1 \frac{\ln(1-z t)}{t}\,dt$ by Gaussian quadrature (9 divisions):

    With[{div = 9},
      With[{x = Sort[formalX /. Solve[LegendreP[div, formalX] == 0, formalX] // N]},
        With[{
          y = Chop[x/2 + 1/2], 
          w = Table[2/((1 - x[[i]]^2)*Derivative[0, 1][LegendreP][div, x[[i]]]^2), 
                {i, 1, div}]},
          With[{
            realRegion2expr =
              -1/4.*Sum[
                      Chop[w[[i]]] Log[1 - 2 y[[i]] var + y[[i]]^2 var^2]/y[[i]], 
                      {i, 1, div}]},
            realRegion2 = Compile[{{var, _Real}}, realRegion2expr]]
         ]
       ]
     ];
    
  3. In region 3, apply the dilogarithm identity $\text{Li}_2(z) = -\underbrace{\text{Li}_2(1-z)}_\text{region I} - \ln(z)\ln(1-z)+\pi^2/6$, where the dilogarithm in the RHS is to be evaluated in region I.

    realRegion3 = Compile[{{x, _Real}}, 
     If[x == 1, Pi^2/6., -realRegion1[1 - x] - Log[x] Log[1 - x] + Pi^2/6.],
       CompilationOptions -> {"InlineExternalDefinitions" -> True}];
    
  4. In region 4, apply the dilogarithm identity $\text{Li}_2(z) = -\underbrace{\text{Li}_2(1/z)}_\text{region I,II,III} - \frac{1}{2}\ln^2(-z)-\pi^2/6$, where the dilogarithm in the RHS is to be appropriately evaluated in region I, II or III depending on the value of $1/z$.

    So first, I need to put together the functions realRegion1, realRegion2 and realRegion3 so that it correctly evaluates on the real line segment $-1 \leq z \leq +1$ appropriately:

    realSegment = 
      Compile[{{x, _Real}},
        If[-0.5 <= x <= 0.5, realRegion1[x],
          If[x <= 0, realRegion2[x], realRegion3[x]]
        ], CompilationOptions -> {"InlineExternalDefinitions" -> True}
      ];
    

The output is quite satisfactory. You can compare Plot[reLi2[x], {x, -5, 5}] with Plot[Re@PolyLog[2, x], {x, -5, 5}].

As an exercise in writing a good Compile function, I want to do the simple task of coding a routine that outputs the real part of the dilogarithm function reLi2[z], given a complex number z as input. I would like some feedback in my code below that does the job.

I am following the strategy outlined in Celestial Mech. Dynam. Astron. 62 (1): 93–98. The idea is to divide the complex $z$-plane into four regions, as I show in the figure below, and to implement a different approximation in each region.

  1. In region 1, approximate the dilogarithm by the defining sum: $\operatorname{Li}_2(z) \approx \sum_{k=1}^\infty x^k/k^2$. Here is the code:

    realRegion1 = Compile[{{x, _Real}}, Sum[x^k/k^2, {k, 1., 23}]];
    
  2. In region 2, approximate the dilogarithm by evaluating the integral $\operatorname{Li}_2(z) \approx -\int_0^1 \frac{\ln(1-z t)}{t}\,dt$ by Gaussian quadrature (9 divisions):

    With[{div = 9},
      With[{x = Sort[formalX /. Solve[LegendreP[div, formalX] == 0, formalX] // N]},
        With[{
          y = Chop[x/2 + 1/2], 
          w = Table[2/((1 - x[[i]]^2)*Derivative[0, 1][LegendreP][div, x[[i]]]^2), 
                {i, 1, div}]},
          With[{
            realRegion2expr =
              -1/4.*Sum[
                      Chop[w[[i]]] Log[1 - 2 y[[i]] var + y[[i]]^2 var^2]/y[[i]], 
                      {i, 1, div}]},
            realRegion2 = Compile[{{var, _Real}}, realRegion2expr]]
         ]
       ]
     ];
    
  3. In region 3, apply the dilogarithm identity $\operatorname{Li}_2(z) = -\underbrace{\operatorname{Li}_2(1-z)}_\text{region I} - \ln(z)\ln(1-z)+\pi^2/6$, where the dilogarithm in the RHS is to be evaluated in region I.

    realRegion3 = Compile[{{x, _Real}}, 
     If[x == 1, Pi^2/6., -realRegion1[1 - x] - Log[x] Log[1 - x] + Pi^2/6.],
       CompilationOptions -> {"InlineExternalDefinitions" -> True}];
    
  4. In region 4, apply the dilogarithm identity $\operatorname{Li}_2(z) = -\underbrace{\operatorname{Li}_2(1/z)}_\text{region I,II,III} - \frac{1}{2}\ln^2(-z)-\pi^2/6$, where the dilogarithm in the RHS is to be appropriately evaluated in region I, II or III depending on the value of $1/z$.

    So first, I need to put together the functions realRegion1, realRegion2 and realRegion3 so that it correctly evaluates on the real line segment $-1 \leq z \leq +1$ appropriately:

    realSegment = 
      Compile[{{x, _Real}},
        If[-0.5 <= x <= 0.5, realRegion1[x],
          If[x <= 0, realRegion2[x], realRegion3[x]]
        ], CompilationOptions -> {"InlineExternalDefinitions" -> True}
      ];
    

The output is quite satisfactory. You can compare Plot[reLi2[x], {x, -5, 5}] with Plot[Re @ PolyLog[2, x], {x, -5, 5}].

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I am following the strategy outlined in Celestial Mech. Dynam. Astron. 62 (1): 93–98Celestial Mech. Dynam. Astron. 62 (1): 93–98. The idea is to divide the complex $z$-plane into four regions, as I show in the figure below, and to implement a different approximation in each region.

I am following the strategy outlined in Celestial Mech. Dynam. Astron. 62 (1): 93–98. The idea is to divide the complex $z$-plane into four regions, as I show in the figure below, and to implement a different approximation in each region.

I am following the strategy outlined in Celestial Mech. Dynam. Astron. 62 (1): 93–98. The idea is to divide the complex $z$-plane into four regions, as I show in the figure below, and to implement a different approximation in each region.

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