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If we are given a constant $c\in $ the extended complex plane, how can we construct a sequence of complex numbers $z_n$ with $|z_n|\to\infty$ such that the corresponding sequence $\tan(z_n)\to c$?

I have thought of writing $\tan z$ in various forms, including $i(e^{-iz}-e^{iz})/e^{-iz}+e^{iz})$ or $\dfrac{\sin x\cos x+i\sinh y \cosh y}{\cos^2 x+ \sinh ^2 y}$ for $z=x+iy$. I can see that there is no definite limit but I cannot be more specific as to construct te sequence described above.

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The tangent function is periodic with period $\pi$ in the direction of the real axis. You just have to find one value $z$ such that $\tan(z)=c$; then the sequence $z_n=z+\frac1n+n\pi$ has the desired properties.

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    @Steven: Quite true.2012-02-21