Re: Complex path integral wrong

*To*: mathgroup at smc.vnet.net*Subject*: [mg131373] Re: Complex path integral wrong*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>*Date*: Tue, 2 Jul 2013 00:45:02 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-outx@smc.vnet.net*Delivered-to*: mathgroup-newsendx@smc.vnet.net*References*: <kqomj6$p76$1@smc.vnet.net> <kqrj4e$4bn$1@smc.vnet.net>

Am Montag, 1. Juli 2013 11:45:50 UTC+2 schrieb Kevin J. McCann: > A followup on my earlier reply. > > > > If I compare the analytical and numerical results from Mathematica, the > > discrepancy occurs on the first leg of the integration > > (1+I -> 1+I R). It is clear that the numerical integration is correct. > > > > Could this be some kind of branch cut issue with the hypergeometric > > function evaluation? > > > > Kevin > > > > On 6/30/2013 3:26 AM, Dr. Wolfgang Hintze wrote: > > > I suspect this is a bug > > > In[361]:= $Version > > > Out[361]= "8.0 for Microsoft Windows (64-bit) (October 7, 2011)" > > > > > > The follwing path integral comes out wrong: > > > > > > R = 3 \[Pi] ; > > > Integrate[Exp[I s]/( > > > Exp[s] - 1 ), {s, 1 + I, 1 + I R, -1 + I R, -1 + I, 1 + I}] // FullSimplify > > > > > > Out[351]= 0 > > > > > > It should have the value > > > > > > In[356]:= (2 \[Pi] I) Residue[Exp[I s]/(Exp[s] - 1 ), {s, 2 \[Pi] I}] > > > > > > Out[356]= (2 \[Pi] I) E^(-2 \[Pi]) > > > > > > Without applying FullSimplify the result of the integration is > > > > > > In[357]:= R = 3*Pi; > > > Integrate[ > > > Exp[I*s]/(Exp[s] - 1), {s, 1 + I, 1 + I*R, -1 + I*R, -1 + I, 1 + I}] > > > > > > Out[358]= > > > I*E^((-1 - I) - 3*Pi)*((-E)*Hypergeometric2F1[I, 1, 1 + I, -(1/E)] + > > > E^(3*Pi)*Hypergeometric2F1[I, 1, 1 + I, E^(-1 + I)]) + > > > I*E^(-I - 3*Pi)*(Hypergeometric2F1[I, 1, 1 + I, -(1/E)] - > > > E^(2*I)*Hypergeometric2F1[I, 1, 1 + I, -E]) + > > > I*E^I*(Hypergeometric2F1[I, 1, 1 + I, -E]/E^(3*Pi) - > > > Hypergeometric2F1[I, 1, 1 + I, E^(1 + I)]/E) + > > > I*E^(-1 - I)*(-Hypergeometric2F1[I, 1, 1 + I, E^(-1 + I)] + > > > E^(2*I)*Hypergeometric2F1[I, 1, 1 + I, E^(1 + I)]) > > > > > > which, numerically, is > > > > > > In[359]:= N[%] > > > > > > Out[359]= -2.7755575615628914*^-17 + 2.7755575615628914*^-17*I > > > > > > i.e. zero. > > > > > > On simpler functions like 1, s and s^2 (instead of Exp[I s]) it works out fine, but not so with e.g. Sin[s] in which case we get 0 again (instead of Sinh[2 \[Pi]]). > > > > > > The integration topic seems to be full of pitfalls in Mathematica... > > > > > > Best regards, > > > Wolfgang > > > Kevin, I think you are right, the branch cut might be the reason. Indeed there seem to be some deeper difficulties with the hypergeometric function 2F1(a,b,c;z) (and/or with branch cuts) in Mathematica. Because this function appears frequently as a result of indefinite integration which Mathematica transforms into definite integrals by taking limits, it deserves a somewhat closer look. The function 2F1[a,b,c,z] as a function of z (if it doesn't reduce to a polynomial in z) has a branch cut in the complex z-plane ranging from +1 to +oo. Let us investigate the behaviour on both side of the cut with an example. We have In[280]:= i0 = Simplify[TrigToExp[Integrate[Sin[s]/(Exp[s] - 1), s]]] Out[280]= 1/2 E^(-I s) (Hypergeometric2F1[-I, 1, 1 - I, E^s] + E^(2 I s) Hypergeometric2F1[I, 1, 1 + I, E^s]) Now, let's take the first part only In[296]:= f = Hypergeometric2F1[-I, 1, 1 - I, E^s]; We now compute the values of f on both side of the cut with different methods 1) replacement with a numerical values close to -0 and +0 In[298]:= f /. {s -> 2 + I*y} /. y -> -0.001 Out[298]= -0.07641223288902461 - 0.0652022010883159*I In[299]:= f /. {s -> 2 + I*y} /. y -> Plus[0.001] Out[299]= -2.6886865517869416 + 5.642501876662279*I 2) Limit to the same numerical values as in 1) This leads to the same results as in 1) In[300]:= Limit[f /. {s -> 2 + I*y}, y -> -0.001] Out[300]= -0.07641223288902478 - 0.06520220108831606*I In[301]:= Limit[f /. {s -> 2 + I*y}, y -> Plus[0.001]] Out[301]= -2.688686551786946 + 5.642501876662289*I 3) Limit to exactly y=0 making the distinction using Direction In[305]:= Limit[f /. {s -> 2 + I*y}, y -> 0, Direction -> -1]; N[%] Out[306]= -0.07649229821994431 - 0.06513712192483992*I In[307]:= Limit[f /. {s -> 2 + I*y}, y -> 0, Direction -> Plus[1]]; N[%] Out[308]= -0.07649229821994431 - 0.06513712192483992*I Obviously, the result here is independent of Direction, and it is (approximately) equal to the approximation from the negative side (y<0). I would consider this indifference with respect to Direction a bug. And this might well be a cause of some of the difficulties in Integrate. I'm not sure because I don't know which limiting procedure is used by Mathematica in order to create definite integrals out of indefinite ones. Maybe others who know more can clarify the situation. PS: By the way, you can easily check that Mathematica behaves correctly with other functions with a similar branch cut, e.g. the function g[z_] := Sqrt[1 - z] at z=2+Iy, so it seems that there is a particular difficulty with the hypergeometric function. Which is especially unpleasant in view of the importance of this type of function in Mathematica's integration algorithm. Best regards, Wolfgang