Questions tagged [complex-numbers]
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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Analogue of roots of unity in n-sphere
The $n$-th roots of unity $z_1,…,z_n$ in $\mathbb{C}\equiv \mathbb{R}^2$ for $n$ prime have an interesting property: for $0\leq p<n$ and $u$ a unit vector, the sum $$\sum_{k=1}^n \langle z_k,u\...
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For a matrix $A(z)$ that represents the operation of multiplication with a complex number $z$, what does it mean for $e^{A(z)t} = A(e^{zt})$?
We can have a complex number $z = a + bi$ that determines a matrix $A(z)$ in the following way:
$$ A(a + bi) = \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $$
This matrix represents the ...
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Polar form of the sum of two complex numbers.
I was trying to express the addition of two complex numbers only in function of its absolute values and arguments while preserving the result in polar form, with the fewest terms possible, for ...
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Explanation for why $(-1)^{-i} = e^\pi$?
I can't find any explanations online as to why the following holds:
$$(-1)^{-i} = e^\pi$$
I assume there's a simple explanation pursuant to the rules of complex number manipulation, but is there any ...
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Introduction to the Binary Tetrahedral group and the 24-cell
Context and introduction
I was playing with complex number sequences $Z_n=r_n\omega^n=u_n+iv_n$ represented in space and realized that it's always possible to associate up to 48 naturally symmetric ...
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factoring complex conjugation
Let $e_1$ and $e_2$ be variables such that
$$e_1e_1 = 1,\quad e_2e_2 = -1,\quad e_1e_2 = i,\quad e_2e_1 = -i.$$
Denote by $(a, b) := ae_1 + be_2$ and $[a, b] := a + bi$.
As usual,
$$[a, b][c, d] = [ac ...
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A simple yet complex proof that I am unable to solve. [closed]
prove that: ((e^(ax ))cos(bx))^n=((sqrt(a^2+b^2))^n)(e^(ax))cos(bx + n.arctan(b/a))
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What are the solutions of $z^2=-1/\overline{z}$ [closed]
I was only able to find the solution $z=-1$, but according to the fundamental theorem of algebra, shouldn't it have two roots?
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Approximating powers of elements on the unit circle
Let $R \subseteq \mathbb{N}$. We say that $R$ is adequate if:
$$\forall n \in \mathbb{N} \; \; \forall \varepsilon > 0 \; \; \forall w \in S^1 \; \; \exists r\in R \; \; |w^n - w^r| < \...
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How to find principal value of the cubic root?
I tried to find principal value for $\sqrt[3]{z}$ , I started from
$$ z=w^3 $$
So
$$ w_1=\sqrt[3]{r} \exp\left(\frac{Arg(z)}{3} i\right)$$
$$ w_2=\sqrt[3]{r} \exp\left(\frac{Arg(z)+2\pi}{3} i\right)$$
...
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Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
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Trying to solve a complex number question
Currently, I have been preparing for ioqm and this question came in a mock of it. Cant figure out whether the question gives sufficient information to solve it....
Here is the statement:
Suppose a, b, ...
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Using the residue theorem to compute two integrals [closed]
Classify the singular points for the function under the integral and using the residue theorem, compute:
(a) $$ \int_{|z-i|=2} \frac{z^2}{z^4 + 8z^2 + 16} \, dz, $$
(b) $$ \int_{|z|=2} \sin\left(\frac{...
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Replacing z with its conjugate to solve questions in complex numbers [closed]
I have noticed that in many questions of Complex numbers , specially those involving a polynomial in z , we replace z with zbar and solve the question, what's the rationale behind doing this. And does ...
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Orthonormal basis for $\mathbb{C}^2$ over $\mathbb{R}$ [closed]
$\mathbb{C}^2$ is a 4-dimensional vector space over $\mathbb{R}$ with basis $\left\{\begin{bmatrix} 1 \\ 0 \end{bmatrix}, \begin{bmatrix} i \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \end{bmatrix}, ...
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