Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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Limit points of a sequence contained in $S^1$

Let $\theta\in (0,2\pi)$ be a real number such that $\displaystyle\frac{\theta}{\pi}\notin\mathbb{Q}$. We define $z:=\cos(\theta)+i\sin(\theta)\in S^1\subseteq\mathbb{C}$ and let ...
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1answer
96 views

Math question complex numbers?

I have to find the roots of $(i)^{1/6}$ ...so I find $k= 0, 1, 2, 3, 4, 5$... the angle is zero degrees apparently...so the first root is $i^{1/6}\times [\cos (0+2\times 0\times \pi)/6 + i\times ...
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1answer
85 views

Finding a basis

Finding the basis for the kernel of: \begin{pmatrix} a & b \\c & d\end{pmatrix} $which$ $maps$ $to:$ \begin{pmatrix} a \\a\\3a + b \end{pmatrix} It's all complex, but I'm not sure if ...
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187 views

Modulus of a complex function: $\Psi(x)=A_0 e^{-kx^2} e^{i\alpha x}$

I am given $$\Psi(x)=A_0 e^{-kx^2} e^{i\alpha x}$$ Here, $\large A_0=[\frac{1}{\pi \sigma_0^2}]^\frac{1}{4}$, $\large k=\frac{1}{2\sigma_0^2}$, and $\large \alpha=\frac{p_0}{\hbar}$ I want to find ...
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1answer
171 views

Complex numbers with modulus argument

I am stuck on this question for homework: "Using the fact that if $z = \cos\theta + i \sin\theta$, then $z^n + z^{-n} = 2\cos n\theta$. Show that $$\cos^3 \theta = \tfrac14(\cos3 ...
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175 views

How to find the formula of a spiral using least-squares(regression)?

Assume data from a plane which are roughly showing a spiral. I want employ the rationale of regression to find the parameters for the best fit by some spiral. That means, I have to estimate the ...
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1answer
31 views

A complex conjugation mulplication

Compute ||a|| given that e/ā=5/4+i/4 and ea=10+2i I found sqrt(13) as answer, but the solution says its 2sqrt(2), am I doing something wrong? Thanks a lot!
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124 views

What is the correct definition for an imaginary number?

The following is taken from Wikipedia's definition. An imaginary number is a number whose square is less than or equal to zero. But I also heard that An imaginary number is a number whose ...
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2answers
587 views

How do we find the integral of this conjugate? [closed]

I'm not sure how to find $$\int_C \overline{z}^2\ dz$$ where $C$ is the circle $|z|=2$ traveled counterclockwise once.
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1answer
103 views

Simplify the complex equation

For the function $$G(w) = \frac{\sqrt2}{2}-\frac{\sqrt2}{2}e^{iw},$$ show that $$G(w) = -\sqrt2ie^{iw/2} \sin(w/2).$$ Ive been told to use the equation below by use of the complex sine ...
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1answer
580 views

Separating equation into real and imaginary parts and eliminating integral

I have the following equation: $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} ...
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1answer
110 views

Arithmetic question regarding $\sqrt{1/-1} = \sqrt{-1/1}$ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? $i^2$ why is it $-1$ when you can show it is $1$? Can someone please point out what I'm doing wrong here? $$ \frac{1}{-1} = ...
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117 views

Diametrically opposite points go to diametrically opposite points under stereographic projection

I asked this question before here but I didn’t get a proper answer. So here I am stating it more clearly : Suppose $P_1$ and $P_2$ are two diametrically opposite points of a circle $C$ in the ...
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119 views

Change in angle between curves due to stereographic projection

Suppose I have say two curves on the complex plane intersecting at a point $P$. Then is the angle between those curves at $P$ same as the angle between their spherical images on the Riemann Sphere (by ...
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98 views

funny complex equality 1 = -1 [duplicate]

Possible Duplicate: $i^2$ why is it $-1$ when you can show it is $1$? Try to find what's wrong there: nb: the squareroot can be defined for all complex numbers as $\exp(1/2\cdot\log(z))$ ...
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1answer
60 views

A set of fixed points

How can we go about finding a Moebius map that fixes the set $\{z_1=x+iy,\,\,\, z_2={1\over iy-x}\}$ for some $x,y\in \mathbb R$ that does not correspond to rotation about any arbitrary axis of the ...
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1answer
243 views

what is difference between the square of an operator and the expectation value of that operator

operator $\hat A$ is a mathematical rule that when applied to a ket $\hat A|\phi\rangle$ transforms it into another ket $\hat A|\phi '\rangle $ and too for bra. $\langle \phi| \hat A|\phi\rangle$ ...
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97 views

Convert triangular real matrix to hermitian

We are developing some computer program which at some point uses a library (for which we do not have access to its source code) to solve the general eigenvalue problem; given two input real symmetric ...
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0answers
91 views

Uniqueness of homography

Let $(z_{1},z_{2},z_{3})$ and $(z'_{1},z'_{2},z'_{3})$ be two $3$-tuples of complex coordinates of non collinear points. How can I prove that there exists a unique homography $h$ such that ...
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2answers
27 views

Find $3^{2-i}$ in the form x+yi

Find $3^{2-i}$ in the form x+yi How do I do this question? $e^{\ln3}$$^{^{2-i}}$ Is that right so far?
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2answers
109 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
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3answers
214 views

What is the value of $\lim_{x\to 0} x^i$?

What is the value of the following limit? $$\lim_{x\to 0} x^i$$ Wolfram Alpha gives an insane result, so does Mathematica.
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2answers
93 views

If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]

If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
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3answers
95 views

A set such that for every $i \in S$, $i^2 \in S$ as well.

I was considering the largest possible set of complex numbers which contained the squares of every element; that is, the largest possible set $S$ such that for every $i \in S$, $i^2 \in S$ as well. ...
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3answers
60 views

complex roots calulation question

How can we find the roots of an equation such as:$z^2 +z +1=0 ,z \in \mathbb{C} $ ?
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2answers
78 views

What does the Cayley table for $+$ in $\mathbb{C}$ look like?

Below is the Caley table for the $*$ operator, but how do I fill in the table for operator $+$? In general, given an operator $*$ acting on a set, $S$, can I turn this into a field by selecting the ...
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2answers
51 views

complex numbers algebraically

solve algebraically $x^2+2ix-5=3i$. My Solution: I tried using quadratic solution ( $a= 1 , b= 2i, c= -5-3i$ ) but it is wrong.
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2answers
119 views

contradicting identity theorem?

the identity theorem for holomorphic functions states: given functions $f$ and $g$ holomorphic on a connected open set $D$, if $f = g$ on some open subset of $D$, then $f = g$ on $D$ Let $f(z) = \sin ...
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1answer
81 views

Is there any meaning on $\displaystyle\Gamma(i)$ [closed]

Is there any meaning on the $\displaystyle\Gamma(i)$ ?
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72 views

Complex Numbers Question

1) let $Z_0$ be a solution of $Z^{13}-13Z^{7}+7Z^{3}-3Z+1=0$, Is it true that $Z_0$'s conjugate is also a solution? 2) let $Z_0$ be a solution of $Z^{2}+iZ+2=0$, Is it true that $Z_0$'s conjugate is ...
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Separating a Complex Valued Function

Is there a formula (with mathematical reasoning) for separating a complex-valued function $f(z)=f(x+iy)$ into the form $ f(z)=u(x,y) + iv(x,y)$? Thank You, C.A
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1answer
101 views

The strange (for me) case of Mod of Iota.

This might be a silly question to some, but I need some help in this topic. Iota, denoted as 'i' is equal to the principal root of -1. Therefore, $\iota^2 = -1$ When studying Modulus, I was ...
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2answers
43 views

Is it appropriate to apply Euclidean Distance to Complex Numbers?

Would complex numbers be considered as part of Euclidean Space? Would this measurement give an accurate result? If not, what would be a more appropriate distance measurement/similarity measure with ...
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1answer
35 views

Determine the real number a so that… [closed]

Determine the real number "a" so that $$ Z= \frac{1+3i}{2-ai} $$ has: $$\arg z = 0 $$
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2answers
37 views

Complex roots problem [duplicate]

I've got a complex equation with 4 roots that I am solving. In my calculations it seems like I am going through hell and back to find these roots (and I'm not even sure I am doing it right) but if I ...
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2answers
45 views

Consider the complex exponential function $f \colon {\mathbb C} \to {\mathbb C}$ given by $f(z) = e^z$.

Describe the image of the vertical line $\text{Re}(z) = 1$ under $f$. How would you solve such question?
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1answer
291 views

Showing that the area of a triangle is $\frac{1}{2}|Im[w_1\bar{w_2}]|$?

I'm reading Beardon's Algebra and Geometry. Let $T$ be a triangle in $\mathbb{C}$ with vertices at $0$, $w_1$, $w_2$. By applying the mapping $z\mapsto \bar{w_2}z$, show that the area of $T$ is ...
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1answer
1k views

How do the roots of unity form a group with respect to multiplication?

How do the roots of unity form a group with respect to multiplication (closure, association, identity) ?
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1answer
45 views

Find all complex solutions to the equation

i) Find all complex solutions to the equation z^4 +1 -i*3^(1/2) = 0 I basically have no clue, any tips/advice/solutions would be great. I could also need some help with another question, this one ...
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1answer
20 views

Draw a set of values in complex plane where the complex number $w=1-3i$ is pure imaginary number.

How would you draw a set of values (in complex plane) where the complex number $w=1-3i$ is pure imaginary number? Could this be the solution? If $Rew=0$.
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2answers
58 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
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1answer
74 views

Complex Number finding the general Value of Theta

If $$(\cos\Theta+i\sin\Theta)(\cos2\Theta+i\sin2\Theta)(\cos3\Theta+i\sin3\Theta) \dots (\cos n\Theta+i\sin n\Theta)=i $$ then show that general Value of $$\Theta=\left[2r+\frac1{n(n+1)}\right]\pi$$ ...
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1answer
23 views

Which of the two points which the function is differentiable

$f(x+iy) = 2x + x^2y + i (7x^2/2 + 7y^2/2 -5y)$ Determine the two points where this function is differentiable. What is the real and imaginary part?
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1answer
63 views

Mapping the line into $f(z)=\sin(z)$ [closed]

Consider the function given as $f(z)=\sin(z)$. We are given a line as $L=\{-\pi/2 + iy | y \in \mathbb R\}$. What is the image of line $L$ under the function $f$? Please Help here.
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1answer
52 views

Set of points $M(z)$

Suppose $z \in \mathbb{C}$ and $a=-1+i$ and $\forall z \in \mathbb{C}\setminus{a}$ $\quad f_a(z)=\dfrac{az}{z-a} $ we suppose in a plan $(P)$: $(D)=\{{M(z) \in (P), ...
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0answers
34 views

Cardinality of the set of complex numbers [duplicate]

Given the continuum hypothesis, does the cardinality of $\mathbb{R}$ ($\aleph_1$) equal the cardinality of the set of complex numbers? If not, what is the cardinality of $\mathbb{C}$? Would it be ...
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2answers
44 views

Finding all the roots from a complex equation

I'm struggling a lot with complex numbers recently. How do I find all the roots for equations like: (1) $\cos z = 3$ (2) $e^{2z} = -e$ (3) $e^z+6e^{-z} = 5$ Thanks
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2answers
37 views

Which is the Hermitian inner product, in terms of conjugate and transpose?

Page 29 of Source 1: Denote the complex conjugate by * : $\mathbf{u \cdot v} = \sum_{1 \le i \le n} u_i^*v_i = (\mathbf{v \cdot u})^*$ Page 1 of Source 2: $\mathbf{u \cdot v} = ...
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2answers
30 views

What does this equation represent in the complex plane? [closed]

What does the set of complex numbers z satisfying the equation $$(3 + 7i) + (10 - 2i) \bar z + 100 = 0$$ represent in the complex plane?
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2answers
50 views

convert complex number in denominator of fraction into polar form

I got this in classroom $$Re\left[\frac{1}{j2\pi f_0 RC + 1}Ae^{j2\pi f_0t}\right]$$ $$=Re\left[\frac{1}{\sqrt{4\pi^2 f_0^2 R^2C^2+1}}e^{-j{tan^-1}2\pi f_0 RC}Ae^{j2\pi f_0t}\right]$$ I attend ...