# [Maxima] dilogarithm float for complex arg?

Edwin Woollett woollett at charter.net
Thu Apr 26 13:27:05 CDT 2012

```An integral which returns an expression in terms
of the dilogarithm function Li[2](z), in which z is
in general complex, is
-------------------------------------------
(%i11) integrate(log(sin(x)),x,1,2);

(%o11) -2*%i*atan(sin(2)/(cos(2)+1))-2*%i*atan(sin(2)/(cos(2)-1))
+2*log(sin(2))-log(2*cos(2)+2)
-log(2-2*cos(2))
+%i*atan(sin(1)/(cos(1)+1))
+%i*atan(sin(1)/(cos(1)-1))-log(sin(1))
+log(2*cos(1)+2)/2+log(2-2*cos(1))/2
+%i*li[2](%e^(2*%i))+%i*li[2](-%e^(2*%i))
-%i*li[2](%e^%i)-%i*li[2](-%e^%i)+3*%i/2
---------------------------------------------
quadpack gives a numerical value -0.0455 for this integral,
but maxima doesn't know how to get numbers for Li[2](z)
if z is complex
------------------------------------
(%i12) float(li[2](1+%i));
(%o12) li[2](%i+1.0)
----------------------------------
wolframalpha gives (0.62 + 1.5 i)  approx
for this float (using PolyLog[2, 1 + I]).

Ted Woollett

```