Question

Find the number of solutions to the equation z^2 + |z|^2 = 0.

Answer 1

Answer on Question #56273 - Math - Complex Analysis

Find the number of solutions to the equation $z^2 + |z|^2 = 0$.

Solution

Put $z = x + iy$. Then the equation becomes

$$(x + iy)^2 + |x + iy|^2 = 0$$ $$x^2 + 2xyi - y^2 + x^2 + y^2 = 0,$$

so equating real and imaginary parts gives

$$\left\{ \begin{array}{l} 2x^2 = 0, \\ 2xyi = 0. \end{array} \right.$$

If

$$2x^2 = 0 \rightarrow x^2 = 0 \rightarrow x = \mathbf{0}.$$

If

$$2xyi = 0 \rightarrow x = \mathbf{0} \text{ or } y = \mathbf{0}$$

Thus,

if $x = 0$, then $y = t$, where $t$ is an arbitrary real constant;

if $y = 0$, then it follows from the first equation of the system that $x = 0$, but this case is a part of the previous condition, when $t = 0$.

So $z = x + iy = 0 + ti = ti, t \in \mathbb{R}$, hence there are infinitely many solutions.

Answer: There are infinitely many solutions $z = ti, t \in \mathbb{R}$.

www.AssignmentExpert.com