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I'm not sure what to do here. It seems like Morera's Theorem, but even then I'm not sure how to choose the triangular path.

bmmcutet12
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    https://math.stackexchange.com/questions/2373844/if-f-mathbbc-to-mathbbc-is-continuous-and-analytic-off-1-1-then-i –  Jul 27 '17 at 18:02
  • First please include the full problem statement in the body of your Question, not merely in the title. Second the problem as posed in the title is confusing. Probably you mean "analytic on the complement of $[-1,1]$" as in the earlier Question noted by @PaulK, but since there is no discussion of that condition in the body of the Question, it is easy for a Reader to be misled. – hardmath Jul 27 '17 at 18:22

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According to Morera, you need to show that $\int_\gamma f(z)\; dz= 0$ for every triangular contour $\gamma$. The only concern is for triangles that intersect $[-1,1]$. The real axis splits such a triangle into two parts. Approximate your contour integral as the integral over a contour in the upper half plane plus an integral over a contour in the lower half plane.

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Robert Israel
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