# Tagged Questions

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

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### Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$

my question is too simple. We know all that if we define the exponential function on $\mathbb{C}$ then we define the real part and imaginary part of $\exp{it}$ as $\cos{t}$ and $\sin{t}$. So if we ...
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### Why is the complex plane shaped like it is?

It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someome ...
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### If $f$ has pole at $0$ then show that $e^f$ can't have pole at $0$.

i am trying to show that if $f$ has a pole at $0$ then $e^f$ can't have removable singularity at $0$ ? I tried to show that but i have a problem . I assume that $e^f$ has removable singularity ...
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### Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{4\pi i}{N}-\frac{2\pi k}{N}\right)\right)=0$?

Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{4\pi i}{N}-\frac{2\pi k}{N}\right)\...
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### Calculating the gcd of complex numbers

I need help in calculating the gcd of complex numbers For Example: $\gcd(3+i,1-i)$. The problem is,I don't even know what's the algorithm for complex numbers...
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### High powers of complex numbers [closed]

I have these two questions that I am trying to solve. I know that I am suppose to use De Movire's Theorem but I am getting stuck. Can you guys please help out? Thanks. Compute the following ...
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### Easy partial fraction decomposition with complex numbers

There is an easy method to perform a partial fraction decomposition - described here, under the "Repeated Real Roots" title, for the coefficient A2. The problem is ...
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### How are the following factors 'linear'.

What does it mean for factors to be linear? Q: Find the four linear factors of: $$z^4+z^3+z^2+z+1$$ I got the following: $$(z-e^{i \pm {2\pi \over 5}} )(z-e^{i \pm {4\pi\over 5}} )$$ I though ...
### Computing $(3+a_1)(3+a_2) \ldots(3+a_n)$ [closed]
If $n+1$ is an odd positive integer and $1,a_1,a_2,\ldots, a_n$ are the $(n+1)^{th}$ roots of unity, then $(3+a_1)(3+a_2) \ldots(3+a_n)=$ ?
### How to simplify $\sqrt{-8}$
How would I go about simplifying square root of $-8$? I know I can rewrite that as $\sqrt{(-1)(8)}$, and then I would get $i\sqrt{8}$, but how do I simplify that $8$ further? Thanks for your help.