Exactly as in the title: is it generally true that $Re(\int_\gamma f(z)dz) = \int_\gamma Re(f(z))dz$. If not, what would be a case in which it is false? I was thinking a counterexample would follow out of a cut line of a principle branch, but I haven't found anything yet.
2026-04-03 17:33:38.1775237618
Real value of a complex contour integral equal to the contour integral of the real value of a complex function?
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The right hand side of your equation isn't even a real number in general.
For an example, integrate the constant function $1$ along a segment not parallel to the real axis.