 Solution

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

Here the zeros accumulate only at the boundary point zero. An explicit holomorphic, nonzero example on the upper half-plane is

f (z) = sin (πi / z) .

z_{n} = i / (n + 1)

its argument is

π (n + 1)

, so it vanishes. It is holomorphic because the only excluded point of its formula is zero; it is not identically zero, for instance at

z = 2 i

. Thus the answer is yes.

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