2
votes
2answers
117 views

Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$, …

Show that if $z = e^{i\theta}$ is a solution to $0 = z^n + a_{n-1}z^{n-1} + \cdots + a_1z + a_0$ [1] where all $a_i$ are real, then $0 = a_{n-1}\sin\theta + a_{n-2}\sin2\theta + \cdots + a_0 \sin ...
-1
votes
3answers
152 views

simplification of an complex exponential equation

There are these steps in a solutions manual I do not follow. I struggle to find any good and problem specific information about this kind of math wizardry on my own. I don't really know what to google ...
2
votes
4answers
91 views

rewrite $2ie^{i\pi}+i^3$

i am asked to rewrite $2ie^{i\pi}+i^3$ into $x+iy$ form. i just tried all what i know so far, but couldnot come to solution. i said: $2ie^{i\pi}+i^3=2ie^{i\pi}-i$ but further i am stuck really. i am ...
18
votes
4answers
763 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
2
votes
2answers
72 views

complex numbers - proof of this statement

i am trying to prove this statement, i dont but how to start. $$\forall z,w \in \mathbb{C}\quad |z|^2+|w|^2=\frac{1}{2}(|z+w|^2+|z-w|^2)$$ can someone please show me how start?
0
votes
1answer
39 views

what is the difference - sorry for over-simplicity

i am asking too simple question, sorry for that. what is the difference between these two imaginär numbers? $\operatorname{Im}(| \sqrt2+3i|^2)$ vs. $\operatorname{Im}((\sqrt2+3i)^2)$ $| ...
2
votes
1answer
59 views

A problem with polynomials.

This is a problem from a test in my course in analytic functions. I didn't manage to solve it. Could you please give me a hint? The problem is: Calculate the third root of the sum of the coefficients ...
1
vote
4answers
273 views

solve complex equation

$x^8 = \frac{1+i}{\sqrt{3} - i} = \frac{\sqrt[8]{\frac{2}{\sqrt{2}}}(\cos \frac{\pi}{4} + i \sin{\frac{\pi}{4}})}{2 \cos \frac{\pi}{6} + i \sin \frac{3\pi}{2}}$ What's the way to solve this kind of ...