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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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Find the $Z$-transform of $\sin (\alpha k), k \ge0$
Find the $Z$-transform of $\sin (\alpha k), k \ge0$ Solution Could anyone please explain how we got the second step (in terms of $e$ and $i$) after writing it in basic $Z$-transform notation? And how ... trigonometry complex-numbers trigonometric-series z-transform- 55
How to perform multiple rotations by direction vectors?
[I'm not sure whether I make the right choice of wording.] I am looking for some performant algorithm for a rotation of multiple points in $\mathbb{R}^2$ (and maybe also $\mathbb{R}^3$ with one ... matrices complex-numbers vectors rotations quaternions- 111
what is the definition of the distance between a complex number with $\mathbb{R}^{+}$?
what is the definition of the distance between a complex number with $\mathbb{R}^{+}$? This is $$ \text{dist}(z,[0,\infty)) \ \ z \in \mathbb{C}? $$ complex-numbers- 151
If $z\in\mathbb C\setminus[0,\infty)$, why $\text{dist }(z, [0,\infty))=:\delta>0$?
If $z\in\mathbb C\setminus[0,\infty)$, why $\text{dist }(z, [0,\infty))=:\delta>0$? complex-numbers- 35
An explicit description of all conjugations on $\mathbb{C}$
The "standard" conjugation for a complex number $a+bi$ is $\overline{(a+bi)}=a-bi$. If we see $\mathbb{C}$ as a one dimensional abelian Lie algebra, we can associate to this conjugation a ... linear-algebra complex-numbers lie-algebras- 389
A field automorphism that is the identity outside a subfield
I was reading Lemma 2 of Daniel Lascar' The group of automorphisms of the field of complex numbers leaving fixed the algebraic numbers is simple whose statement is the following: Assume $g \in \text{... abstract-algebra complex-numbers field-theory automorphism-group- 2,113
Show that a) $\frac{1}{\frac{1}{z}}=z$. b) Show that if $\vert z \vert \lt 1$ then $ \vert Re (2 + \bar z + z^3)\vert \le 4$.
Show that a) $\frac{1}{\frac{1}{z}}=z$. b) Show that if $\vert z \vert \lt 1$ then $ \vert Re (2 + \bar z + z^3)\vert \le 4$. My attempt: a) $\frac{1}{\frac{1}{z}}= \frac{1}{z^{-1}} = (z^{-1})^{-1} = ... complex-numbers solution-verification- 11
Evaluate $\int_{-\infty}^\infty e^{i(\omega-\omega_0)t}\mathrm\;{dt}$
Evaluate $$\int_{-\infty}^\infty e^{i(\omega-\omega_0)t}\mathrm\;{dt}$$ This is an integration from a textbook chapter discussing wave involving Fourier transform, which try to deal with dispersion. ... integration definite-integrals complex-numbers fourier-transform- 1
Can we construct the complex numbers by using $i^2 = -1$ as an axiom?
Informally we can construct the complex numbers by starting with the real numbers and adding a new element $i$ which satisfy $i^2 = -1$. If we want to formalize this construction we usually use ... abstract-algebra complex-numbers field-theory- 343
Understanding why the angle between lines $a_1 + tb_1, a_2 + tb_2, a_i, b_i \in \mathbb{C}, i = 1, 2$ is $\mathrm{arg}(b_2/b_1)$
Suppose that $b_1 \neq 0$ and $l_1(t) = a_1 + tb_1, l_2(t) = a_2 + tb_2, a_i, b_i \in \mathbb{C}, i = 1, 2$ are two complex lines. My CA book states that the angle between the two lines is given by $\... complex-analysis geometry complex-numbers analytic-geometry- 239
Cramer's rule and centre of a circle
The exact problem of finding the centre of a circle circumscribing the triangle formed by three complex numbers $a_1, a_2$ and $a_3$ is considered in detail here: Finding center and radius of ... linear-algebra complex-numbers systems-of-equations- 1,521
Conditions of equilateral triangle in complex plane/equivalence of angles between segments
Suppose that $z_1, z_2, z_3 \in \mathbb{C}$ such that $z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_1z_3 + z_2z_3$. In order to conclude that the triangle formed by $z_1, z_2, z_3$ is equilateral, it is ... geometry complex-numbers analytic-geometry- 1,521
Can any domain in the Riemann sphere be approximated by finitely connected Jordan domains?
Is it true that for any domain $\Omega \subset \bar{\mathbb{C}}$ we can find finitely connected Jordan domains $\Omega_n$ such that $$ \Omega_n \subset \bar{\Omega}_n \subset \Omega_{n+1} \subset \... general-topology complex-numbers connectedness- 527
$G \subset e^{\frac{2kπ} {n} i}, k \in \mathbb{Z}$. Show that $G$ is a group under multiplication
Let $G$ be the subset of complex numbers of the form $e^{\frac{2k\pi} {n} i}, n,k \in \mathbb{Z+}$. Show that $G$ is a group under multiplication. How many elements does $G$ have? Associativity under ... group-theory complex-numbers solution-verification- 3,623
Powers of roots of unity are also roots of unity?
So I was thinking about roots of unity, loosely inspired by this video (17:28 ff.), in which the author, as best I understand it, says: Suppose $z$ is a 5th root of unity. So, by definition, $z^5=1$. ... complex-numbers roots-of-unity- 299
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