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numerics - Symbolic integration in the complex plane


Context


While answering this question, I defined (symbolic and numerical) path integrations as follows



 ContourIntegrate[f_, par : (z_ -> g_), {t_, a_, b_}] := 
       Integrate[Evaluate[(f /. par) D[g, t]], {t, a, b}]

 NContourIntegrate[f_, par : (z_ -> g_), {t_, a_, b_}] := 
       NIntegrate[Evaluate[D[g, t] (f /. par) /. t -> t1], {t1, a, b}]

I also defined a piecewise contour


Clear[pw]; 
pw[t_, a_: 1] = Piecewise[{{a Exp[I t], t < Pi}, {-a + 2 a (t - Pi)/Pi, t >= Pi}}]
ParametricPlot[pw[t] // {Re[#], Im[#]} &, {t, 0, 2 Pi}]


Mathematica graphics


While checking these routines on wikipedia examples, I tried numerically


 Table[NContourIntegrate[Exp[I i x]/(x^2 + 1), x -> pw[t, 2], {t, 0, 2 Pi}], {i, 2, 5}] // Chop

(* {0.425168,0.156411,0.0575403,0.0211679} *)

which corresponds accurately to (see example II for Cauchy distributions)


Table[Exp[-i] Pi, {i, 2, 5}] // N


On the other hand, the symbolic integration


ContourIntegrate[Exp[I x]/(x^2 + 1), x -> pw[t, 2], {t, 0, 2 Pi}] // FullSimplify

returns 0.


Question


What am I doing wrong?


Attempts


Example I and III work ;-)


ContourIntegrate[1/(x^2 + 1)^2, x -> pw[ t, 2], {t, 0, 2 Pi}] // FullSimplify
NContourIntegrate[1/(x^2 + 1)^2, x -> pw[ t, 2], {t, 0, 2 Pi}] // Chop


(* Pi/2  1.5708 *)

and


NContourIntegrate[1/I/x/(1 + 3 ((x + 1/x)/2)^2), x -> Exp[I t], {t, 0, 2 Pi}]//Chop
ContourIntegrate[1/I/x/(1 + 3 ((x + 1/x)/2)^2), x -> Exp[I t], {t, 0, 2 Pi}]

(* 3.14159  Pi *)

Answer



The problem lies with the fact that my init.m file has



 SetOptions[Integrate, GenerateConditions -> False]

If I use


 SetOptions[Integrate, GenerateConditions -> True]

or define


 ContourIntegrate[f_, par : (z_ -> g_), {t_, a_, b_}] := 
   Integrate[Evaluate[(f /. par) D[g, t]], {t, a, b}, GenerateConditions -> True]

the discrepency vanishes.



I guess this makes this question too narrow to be of general interest!


In any case, the bring home message is don't do complex integration without paying attention to branch-cuts !


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