Fake roots of a Complex Equation.

Question

I am looking for the roots of the complex function $f(x)=1+(ix)^N, N\in R$,

$\Rightarrow 1+(ix)^N= 0$

$\Rightarrow (ix)^N=-1$

$\Rightarrow ix=(-1)^{1/N}$

$\Rightarrow ix=(e^{i(2k+1)\pi})^{1/N} , k=0,1,2,3.....$

Roots are given as, $ ~x=-i e^{i(2k+1)\pi /N }, k=0,1,2,3.....$.

Now first four roots $(k=0,1,2,3)$ for $N=3.6$ are given as: {$0.766044 - 0.642788 i,0.5 + 0.866025 i, -0.939693 + 0.34202 i,$ and $ -0.173648 - 0.984808i$}.

Now, if I put back them into $f(x)$, then -

$$\{0,~0,~ 1.80902 - 0.587785 i,~ 1.80902 - 0.587785 i \}$$

So only two are genuine roots, rest two are fake.

Now the Ques is: what/where I am missing for getting the fake roots?

Answer 1

The procedure you have used cannot be correct because

• you compute some complex number of the form $x^{i\theta}$ where $\theta$ can exceed $2\pi$;

• you express it in Cartesian form, hence argument information is lost.

• you take the exponential.

So you are not computing

$$e^{i N\theta}$$ but $$e^{iN(\theta\bmod 2\pi)}.$$