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Let $\epsilon $ be a positive real number and $a$ a complex number.

Prove the function $f(z)= \sin z +\frac{1}{z-a}$ has infinitely many zeros in the strip $|\mathrm{Im}z| < \epsilon.$

Thanks in advance!

  • 2
    This is an old qualifying exam problem from Rice: [Question 6 on page 6](http://math.rice.edu/~idu/Sp05/AnalysisV11.pdf).2012-08-19

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