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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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up vote 4 down vote accepted

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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Thank you, joriki! I can't believe I forgot about the periodicity of tan! – help pleeeze Feb 21 '12 at 21:42
Strictly speaking, the sequence $z_n = z+n\pi$ also has the desired properties - it just achieves $c$ at every member, rather than approaching it in the limit, but that doesn't change that $\tan(z_n)\rightarrow c$. – Steven Stadnicki Feb 21 '12 at 22:07
@Steven: Quite true. – joriki Feb 21 '12 at 22:26

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