Complex Analysis: Trig Functions

**joestevens** · October 27th 2010, 04:00 PM

Find all solutions to:
1) $cosh(z)=-1$ -----done
2) $cos^2 (z)=4$
3) $tan(z)=i$

**TheEmptySet** · October 27th 2010, 04:11 PM

Remember that

$\cos(z)=\frac{e^{iz}+e^{-iz}}{2}$

This gives

$\left(\frac{e^{iz}+e^{-iz}}{2} \right)^2=4 \iff e^{2iz}+2+e^{-2iz}=16$

$(e^{2iz})^2-14e^{2iz}+1=0$

This is quadratic in $e^{2iz}$

Can you finish from here?

You will use a similar trick for tangent just write it as exponentials.

**joestevens** · October 29th 2010, 09:09 AM

I finished from there.

Thanks.

I got (3) to be no solution.

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