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Question: Semi-incorrect complex symbolic integration

Question:Semi-incorrect complex symbolic integration

Johan Persson 68 Maple
complex integration
January 18 2010
0
(Using Maple 13.01) When performing a symbolic complex integration where the integral will have two (or more) distinct values depending on a parameter in the function maple sometimes return the correct indication of branches and for more complicated expressions the integration only gives the result of one branch (not indicating the different possibilities) A trivial example to help explain my question/thought f(z) = 1/(z+2) Path of integration, |z+1|=r, this means that for |r| <> 1 the integral will simply be 2*Pi*i If I in maple evaluate this as int(subs(z=-1+r*exp(I*t), 1/(z+2)*I*r*exp(I*t)), t=0..2Pi) assuming r>0 this will give two (incomplete) branches as piecewise(r=1, undefined, 0) where the more complete (correct) result should be piecewise(r=1, undefined, r>1, 2*Pi*I, 0) But this is not the only issue. For the slightly more complicated denominator f(z) = 1/(z^3+2) With the same path as above maple will (with/without assumption) always evaluate this to 0 not indicating the possibility of impact of the integration radius with the other possible value of 2^(1/3)*I*Pi/3 While the above two examples are trivial I came across this issue in some more complicated expressions where it wasn't so obvious where the singularities was and the impact of (several) parameters. Question: 1) Is this simply a limitation in the symbolic integration or is there some better way of doing these types of integration (to make sure one becomes aware of the parameter impact)? 2) How does maple determine what branch to evaluate when it "ignores" the parameter, i.e. why did maple chose the 0 result above instead of the alternative 2^(1/3)*I*Pi/3 /J
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