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Suppose one has function $f(z)$ analytic in the unit disk. Suppose closed loop $L$ lies in the disk except for one point $P$ on the boundary. Then the Cauchy integral theorem generally does not apply and one can't always conclude that $\int_L f(z)dz=0$.

Do there exist theorems which do recover the conclusion subject to suitable restrictions on the boundary behavior of $L$ and $f$? How about, for example, for modular forms $f$ and loops $L$ with $P$ sitting on a sufficiently sharp cusp (sharp enough, say so that the two branches of $L$ extending from $P$ lie close together in the Poincare metric)?

Perhaps the right statement might not apply to all $L$, but, say, only to almost all $L$ equivalent up to rotation (again with $P$ on a sufficiently sharp cusp or whatever)?

Please don't hesitate to answer because I haven't get the question exactly right.

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That is pretty much the same as changing the contour of integration when it escapes to infinity. What is normally done is to check that the integral converges and to create a sequence of "shortcuts" for which you can show that the integrals over them are small (usually by the crude length times maximum bound). It is important to understand that you need just a sequence of shortcuts escaping to P, not all of them, and it is possible that some may be better than the others in general. It is not an exact science, of course, but all "general theorems" of this sort I know are based on this idea. – fedja Nov 22 '11 at 14:13

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