For z = x+iy find the relationship between x and y

[mardybum9182]

Homework Statement

For z = x + iy find the relationship between x and y so that (Imz^2)/z^2=-i.

Relevant Equations

modulus

(x+iy)^2 = x^2 + i2xy - y^2

 

[PhDeezNutz]

Are you familiar with "polar form" of a complex number? i.e. Euler's Identity? Using this approach my result is that $x=y$

I'll get you started.

Here are some basic formulas. You can derive them if you want or you can just accept them for the time being.

$z = re^{i \theta} = r \cos \theta + i r \sin \theta$

$\text{Im} \left(z\right) = \frac{re^{i \theta} - re^{-i \theta}}{2i}$

$z^2 = r^2 e^{i 2 \theta}$

$\text{Im} \left( z^2\right) =$?

$\frac{\text{Im} \left(z^2 \right)}{z^2} =$? (You should get $-i$)

Equate the real parts to the real parts, the imaginary parts to the imaginary parts and go from there.

You should get $\theta = \frac{\pi}{4}$

 

[anuttarasammyak]

(x+iy)^2 = x^2 + i2xy - y^2

=Rez^2+i Imz^2, so you know both numerator and denominator.