# Tagged Questions

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

1answer
64 views

### $z= \frac{u-\overline{u}v}{1-v}$ is real is equivalent to $|v|=1$.

Let $u,v$ be complex numbers such as $u,v\notin \mathbb{R}$, and : $$z= \frac{u-\overline{u}v}{1-v}$$ Prove that : $z\in\mathbb{R} \Longleftrightarrow |v|=1$.
2answers
236 views

### Prove $i\notin \mathbb R$

Prove that $\imath$ (defined by $\imath^2=-1$) does not have a position on the Real number line. That is, show that there does not exist two real numbers $a$ and $b$ such that $a<\imath<b$. (I'...
2answers
119 views

### A question on complex numbers

We are given If $\cos(a+ib)$=$r (\cos\theta +i\sin\theta)$ then prove that $e^{2b} = \sin(a-\theta)/­\sin(a+\theta)$ I just tried and got $b = 0$ such that $\cos(a) = ra$. Will there be other ...
3answers
224 views

### Non-existence of a certain holomorphic function on the unit disk

I am trying to prove the following: Let $n\in \mathbb{N}$. Prove that $\not \exists$ a holomorphic function $f$ on the open unit disk satisfying: $f\left(\displaystyle \frac{1}{n}\right) = 2^{-n}$ ...
5answers
175 views

### simplify $(-2 + 2\sqrt3i)^{\frac{3}{2}}$?

How can I simplify $(-2 + 2\sqrt3i)^{\frac{3}{2}}$ to rectangular form $z = a+bi$? (Note: Wolfram Alpha says the answer is $z=-8$. My professor says the answer is $z=\pm8$.) I've tried to figure ...
2answers
276 views

### Euler's formula and $i^x = \cos(x \cdot \frac{\pi}{2})$

While playing around with a plotting software, i just found out that $$f(x) = i^x = \cos(x·\frac{\pi}{2})$$ How does this connect to Euler's formula? Obviously, here, the alternating sign change ...
1answer
73 views

2answers
144 views

### the power sum can completely determine the complex numbers self

If $a_1,a_2,\dots,a_n,b_1,b_2,\dots,b_n$ are complex numbers in $\mathbb C$, and for every $j\in\mathbb N$, we have the power sums $$\sum_{i=1}^n a_i^j=\sum_{i=1}^n b_i^j$$ I want, without applying ...
3answers
1k views

### A square root of i with negative imaginary part

In an ODE class, one assignment question says find the “rectangular” expression z = a + bi (with a and b real) and the “polar” expression |z|, Arg(z) where z is "a square root of i with negative ...
2answers
1k views

### Raising a square matrix to the k'th power: From real through complex to real again - how does the last step work?

I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!) On page 69 it gives an example where a real, square matrix $A=[(a,-b),(b,a)]$ is raised ...
1answer
80 views