 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2008-printed-variant/6/iii/solution

The domain has two connected components. The zeros force the function to vanish on the upper half-plane but impose nothing on the lower one. Define

f (z) = {0, 1, Im z > 0, Im z < 0.

This function is locally constant and hence holomorphic on the stated open set, and is not identically zero on that whole set. Thus the answer is yes. Connectedness is exactly the missing hypothesis for a global identity conclusion.

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