# Thread: im & re as functions of x & y - complex analysis

1. ## im & re as functions of x & y - complex analysis

Find the real and imaginary parts of the following functions as functions of $\displaystyle x$ and $\displaystyle y$.

(i) $\displaystyle z^3$
(iii) $\displaystyle (z+z^{-1}) \ (z\ne 0)$
(vi) $\displaystyle {1\over 1-z} \ (z\ne 1)$

2. Originally Posted by JJMC89
Find the real and imaginary parts of the following functions as functions of $\displaystyle x$ and $\displaystyle y$.

(i) $\displaystyle z^3$
(iii) $\displaystyle (z+z^{-1}) \ (z\ne 0)$
(vi) $\displaystyle {1\over 1-z} \ (z\ne 1)$
1° $\displaystyle (x+iy)^3$ solve and group numbers with "i" and without it so those with "i" will be imaginary ones ...

2° $\displaystyle \displaystyle (x+iy) + \frac {1}{x+iy}$ add those than multiply with $\displaystyle \displaystyle \frac {x-iy}{x-iy}$ and than group Re and Im

6°$\displaystyle \displaystyle \frac {1}{1-x - iy }$ this one multiply with $\displaystyle \displaystyle \frac {1-x+iy}{1-x+iy}$ than do same thing

3. Note that $\displaystyle \text{Re}(z+w)= \text{Re}(z)+ \text{Re}(w).$

So $\displaystyle \text{Re}(z+z^{-1})=x+\frac{x}{x^2+y^2}.$