Originally Posted by

**summerBoy** In my exercise book I couldn't find a way to solve what I've typed below (I know for final solutions anyway):

PROBLEM 1

Find Re(w) and Im(w) [where $\displaystyle z=x+yi$]:

$\displaystyle w=z+\frac{1}{z}$

So we have to write a complex number in a standard form and then extract a real and an imaginary part from there. We work here with the unknown variables ($\displaystyle x $ and $\displaystyle y $).

My way would be:

$\displaystyle w=z+\frac{1}{z}=x+yi+\frac{1}{x+yi}=\frac{(x+yi)^{ 2}+1}{x+yi}=\frac{((x+yi)^{2}+1)(x-yi)}{(x+yi)(x-yi)}=$

$\displaystyle \frac{((x+yi)^{2}+1)(x-yi)}{x^{2}+y^{2}}=...$

If I expand the expression further (i.e. numerator) I don't find a way to write it in a standard form.

However, the final solution is:

$\displaystyle Re(w)=x+\frac{x}{x^{2}+y^{2}}$ and $\displaystyle Im(w)=y-\frac{y}{x^{2}+y^{2}}$

Does anybody know the steps leading to the final solution?

PROBLEM 2

Find all solutions to the equation [where $\displaystyle z=x+yi$]:

$\displaystyle 2z^{2}-3\overline{z}^{2}=10i$

It is a quadratic equation and has a conjugate of complex number in it.

Thanks for help in advance.