FeynCalc manual (development version)

Hill

Hill[x, y] gives the Hill identity with arguments x and y. The returned object is 0.

See also

Overview, SimplifyPolyLog.

Examples

Hill[a, b] 
 
% /. a :> .123 /. b :> .656 // Chop

Li2(1a1b)+Li2(ba)Li2((1a)ba(1b))+log(a)(log(1a)log(1b))+log(1a1b)(log(aba)+log(ab1b)log(a)+log(1b))(log(aba)+log(aba(1b))+log(1b))log((1a)ba(1b))+Li2(a)Li2(b)π26\text{Li}_2\left(\frac{1-a}{1-b}\right)+\text{Li}_2\left(\frac{b}{a}\right)-\text{Li}_2\left(\frac{(1-a) b}{a (1-b)}\right)+\log (a) (\log (1-a)-\log (1-b))+\log \left(\frac{1-a}{1-b}\right) \left(-\log \left(\frac{a-b}{a}\right)+\log \left(\frac{a-b}{1-b}\right)-\log (a)+\log (1-b)\right)-\left(-\log \left(\frac{a-b}{a}\right)+\log \left(\frac{a-b}{a (1-b)}\right)+\log (1-b)\right) \log \left(\frac{(1-a) b}{a (1-b)}\right)+\text{Li}_2(a)-\text{Li}_2(b)-\frac{\pi ^2}{6}

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Hill[x, x y] // PowerExpand // SimplifyPolyLog // Expand 
 
% /. x :> .34 /. y -> .6 // N // Chop 
  
 

ζ(2)Li2(xy)+Li2(1x1xy)Li2((1x)y1xy)Li2(1x)Li2(1y)log(x)log(1xy)log(1y)log(y)\zeta (2)-\text{Li}_2(x y)+\text{Li}_2\left(\frac{1-x}{1-x y}\right)-\text{Li}_2\left(\frac{(1-x) y}{1-x y}\right)-\text{Li}_2(1-x)-\text{Li}_2(1-y)-\log (x) \log (1-x y)-\log (1-y) \log (y)

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