# 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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### Eigendecomposition of a complex symmetric matrix and the same with a shift on the diagonal

Suppose that the matrix $A\in \mathbb{C}^{n\times n}$ is complex symmetric matrix, thus is fulfills $A=A^T$ but also $A\neq A^H$, thus the matrix is not hermitian, but complex symmetric. Let's ...
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### Why is the MacLaurin series proof for eulers formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ valid?

The proof for this $$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the ...
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### How to simplify $\Re\left[\sqrt2 \tan^{-1} {x\over \sqrt i}\right]$?

While solving $$\int \frac{x^2+1}{x^4+1}\,dx,$$ I tried to use partial fractions in the denominator by writing $x^4+1=(x^2+i)(x^2-i)$ And then I got $\Re\left[\sqrt2 \tan^{-1}{x\over \sqrt i}\right]$. ...
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### Using the picture write down all values of $\sqrt[12]{z}$ and then find the main value of that number $z$.

Problem: Let $1+i\sqrt 3$ be one root of $\sqrt[12]{z}$. Display that number in complex plane and then, in that plane, display other roots of number $\sqrt[12]{z}$. Using the picture write down all ...
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### Where does this equation come from? [duplicate]

Since I study 3 years i ask myself very often where does this equation come from? $$e^{i\theta} = \cos(\theta)+i \sin(\theta)$$ Is it found by series expansion?
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### Real Numbers Raised to Imaginary Powers? [closed]

What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples ...
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### locus of complex number 2

Que: If $arg(\frac{z-z_1}{z-z_2})=\pi$ then what is the locus of $z?$ Doubt In my textbook it is written that it represents the straight line joining $A(Z_1)$ and $B(Z_2)$ but excluding the ...
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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 $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?

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{2\pi i}{N}-\frac{\pi k}{N}\right)\...
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### Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right)$$ where I taken ...
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### Triangle inequality for complex numbers

I just start to learn about complex numbers and I want to prove the triangle inequality, which says that if $z$ and $w$ are complex numbers, then $\displaystyle |z + w| \le |z| + |w|.$ My ...
$$\overline{z-2+4i} = 2z+3+8i$$ I got this question on my online assignment. I got to a point where I couldn't get rid of the conjugate of z and I don't know how to expand or what to do with it. I ...