[SOLVED] im & re as functions of x & y

im & re as functions of x & y - complex analysis

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

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

Quote:

Originally Posted by JJMC89

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

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

1° $(x+iy)^3$ solve and group numbers with "i" and without it :D so those with "i" will be imaginary ones ... :D:D:D

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

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

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

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