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I am trying to understand how Mathematica handles the Integral with complex limits. 

NIntegrate[Exp[Sin[y]], {y, I, 2}]

How does NIntegrate works for this limits?

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  • $\begingroup$ This question is closely related. Read the both answers in case of any doubts, namely you can choose any compact curve starting from I and ending in 2. $\endgroup$ Commented Feb 7, 2017 at 12:40

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You can use EvaluationMonitor to see what points were sampled:

{res,data}=Reap[NIntegrate[Exp[Sin[y]],{y,I,2},EvaluationMonitor:>Sow[y]]];

Here is a plot of the data showing that NIntegrate uses a straight line contour:

enter image description here

Graphics[{PointSize[Large],Red,Point@@ReIm@data},Axes->True,AxesLabel->{Re,Im}]

Compare this to choosing a different contour:

{res,data}=Reap[NIntegrate[Exp[Sin[y]], {y, I, I+2, 2}, EvaluationMonitor:>Sow[y]]]; Graphics[{PointSize[Large],Red,Point@@ReIm@data},Axes->True,AxesLabel->{Re,Im}]

enter image description here

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