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

\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}

0

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

\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)

0