Give a discusion of the mapping $f(z)= \frac{1}{2}(z+1/z)$
This is a exercise from functions of one complex variable.
The problem is that I guess I have to draw it or show the propierties about it, I saw its derivate but I don't find anything helpful.
Thanks for your help.
On
May be your function was slightly wrong and you meant $g(z)=\frac{1}{2}(z+\bar{z}^{-1})$ instead.
Then a way to address this discussion, would be to observe that $\bar{z}^{-1}$ is nothing but the image of the point $z$, when inverted at the circle of unity.
Thus the function $g(z)$ provides for any point $z\in\mathbb{C}^*$ the medium between the according image and pre-image.
--- rk
Consider $z=a+bi$. Then you surely know that $\bar{z}=a-bi$.
Moreover $z\cdot\bar{z}=|z|^2=a^2+b^2$. Thus $\frac{1}{z}=\frac{\bar{z}}{|z|^2}$.
And therefore $2 f(z)=z+\frac{1}{z}=a+bi+\frac{a-bi}{a^2+b^2}=a\frac{a^2+b^2+1}{a^2+b^2}+b\frac{a^2+b^2-1}{a^2+b^2}i$.
Or stated differently: $2 f(z)=Re(z)\frac{|z|^2+1}{|z|^2}+Im(z)\frac{|z|^2-1}{|z|^2}i$.
--- rk