Tagged Questions

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

51 views

22 views

Number of roots of a complex exponent

There are $p$ solutions to $\sqrt[\frac{p}q]1$, if $\frac{p}q$ is a fraction in lowest terms. I have found on this website that an irrational exponent has infinite roots. But what about $\sqrt[a+bi]1$...
47 views

How to find the General expression of $\sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k}$ [duplicate]

Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity $$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$ Need help ...
75 views

$e^{a 2\pi i} = (e^{2\pi i})^a$.

When $a$ is any real number , Is it true $e^{a 2\pi i} = (e^{2\pi i})^a$ ? The reason why I ask this question is that I met this situation wheter this equality hold in Calculating Integral in Complex ...
58 views

Complex numbers inside determinant

Let $\begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y$, then what are the values of $x$ and $y$?
173 views

Is there a way to prove that i²=-1? [duplicate]

I have 4 questions regarding the imaginary and complex numbers. (And some ideas) My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe ...
17 views

Complex Conjugate roots with non real coefficients

I understand that a polynomial with real coefficients must have complex conjugate roots (if complex roots exist) Is it possible for a polynomial with non-real coefficients to have complex conjugate ...
36 views

Visualizing a complex function

Ever since I learned about complex valued functions I've been wondering if there was a better visualization for them. Obviously we can't visualize four dimensions, but I was wondering if it would be ...
78 views

Solve in $\mathbb{C}$ : $|z-i| = |z-1|$

I just had that question in my final exam Solve in $\mathbb{C}$ : $|z-i| = |z-1|$ and I couldn't do it. I found a similar thread here : Showing that $\{z\in\mathbb{C}:|z-1|<|z+i|\}$ is an open ...
27 views

Bounded real parts of the solutions of an equation

I'd like to show that, with $a,b>0$, the real parts of the solutions $z_n$ of the equation $$az+\sqrt{z^2-ib}\tanh\sqrt{z^2-ib}=0$$ are bounded. An indication for that can be found if we ...
29 views

94 views

Why is De Moivre's theorem not generalised for $(\sin x+i\cos x)$?

A representation of the form $(\sin x+i\cos x)^n$ can be reduced as follows $$( \sin x + i \cos x )^n$$ $$( \cos (90-x) + i \sin(90-x) )^n$$ $$( \cos (90n - nx) + i \sin(90n - nx) )$$ Now for all ...
66 views

Linear algebra : Solving $i \cdot\bar{z} = 2 +2i$

$i\cdot\bar{z} = 2+2i$ I know that $\bar{z} = a-bi$ so then i get $i(a-bi)=2+2i$ Then $ai+b=2+2i$ (because $i^2=-1$) When 2 complex numbers are equal you usually can equal their parts Ex: $2+2i=a+bi$...