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

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3
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2answers
493 views

Roots of unity?

The $n$th roots of unity are the complex numbers: $1,w,w^2,...,w^{n-1}$, where $w = e^{\frac{2\pi i} {n}}$. If $n$ is even: The $n$th roots are plus-minus paired, $w^{\frac{n}{2}+j} = ...
5
votes
2answers
225 views

Number of Complex Roots of a Complex Polynomial

This is related to the question I asked regarding finding the complex roots of $z^3+\bar{z}=0$. It turned out that there were 5 complex roots, but because the equation was of degree 3 I was only ...
1
vote
2answers
175 views

An inequality for two complex numbers

I recently saw the following inequality for complex numbers: If $a,b\in\mathbb C$ and $|a + b|$ and $|a-b|$ are each less than or equal to 1, then $$|a| + |b^2|/2 \leq 1.$$ How can one prove this?
10
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6answers
661 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
2
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1answer
123 views

What representation should I choose for numerical computation of hypergeometric function ${}_2 F_1(1+i\eta, 2; 2+i\eta; x)$ where $|x|=1$

I have a task - to plot graphics of the function: $$ I(E) = \frac{16i \pi k \mu}{(\beta - ik)^{4}} \frac{1}{1 + i\eta} {}_2 F_1(1+i\eta, 2; 2 + i \eta; x) $$ where $$ x = \left( \frac{\beta + ...
0
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1answer
109 views

Magnitude of a complex number

How might we show that $\Big|{b^2+d^2-a^2-c^2+i2ab+i2cd\over a^2+b^2+c^2+d^2+2}\Big|\le 1$ if we are given that $ad-bc=1$ and $a,b,c,d$ are real?
4
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3answers
147 views

Show that $\forall n \in \mathbb{N} \left ( \left [(2+i)^n + (2-i)^n \right ]\in \mathbb{R} \right )$

Show that $\forall n \in \mathbb{N} \left ( \left [(2+i)^n + (2-i)^n \right ]\in \mathbb{R} \right )$ My Trig is really rusty and weak so I don't understand the given answer: $(2+i)^n + (2-i)^n $ ...
5
votes
2answers
4k views

Multiplying complex numbers in polar form?

Could someone explain why you multiply the lengths and add the angles when multiplying polar coordinates? I tried multiplying the polar forms ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot ...
0
votes
3answers
575 views

Rectangular form of a complex number?

Why does rectangular form serve as an accurate description of a complex number? Why not $a * bi$(multiplication) or another operation? Why does addition describe the relationship between the real part ...
2
votes
1answer
88 views

How to integrate $\int_{\gamma_1} \frac{dz}{z(z-i)}$ with $\gamma_1 = Re^{it}$, $R>1$?

I am stuck calculating the integral $$\int_{\gamma_1} \frac{dz}{z(z-i)}$$ over $\gamma_1 = Re^{it}, R>1$. If I had to integrate over $\gamma_2 = re^{it}, r < 1$, I could just expand the ...
2
votes
2answers
124 views

Need a hint with this question

I'm looking over one of my past papers and I'm having some trouble with the following question. By considering the series expansion of: $\ln(1-z)$, where $z=\frac{e^{i\theta}}{2}$, show that ...
2
votes
2answers
100 views

Solutions to $z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2 = 0, b\in \mathbb R, z \in \mathbb C$

$z_1 = 1+i$ is a given solution. I guess what I have to find is $z_2$ and $z_3$ in $(z - (1 + i))(z - z_2)(z-z_3) = z^3 - (b+6) z^2 + 8 b^2 z - 7+b^2$. I tried to divide the polynomial by $(z - (1 ...
14
votes
3answers
7k views

Is the square root of a negative number defined?

I have been in a debate over 9gag with this new comic: "The Origins" And I thought, "haha, that's funny, because I know $i = \sqrt{-1}$". And then, this comment cast a doubt: There is no such ...
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0answers
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 $\exp(1/2\cdot\log(z))$ ...
2
votes
1answer
193 views

Prove that there is an unique $z$ s.t. $f(z) = z$ where $z$ is a complex number

Let $f$ be analytic on the closed unit disk centered at the origin and $|f(z)| < 1$ for $|z| = 1$. Show that $f$ has exactly one fixed point inside the open unit disk. That is, there exists a ...
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1answer
163 views

A way to define the imaginary axis

There are some ways to define the imaginary axis.Some are obvious, like $\space Re(z)=0 \space$ others not. I set up a condition that I think defines the imaginary axis. Let $x$ be a real number, ...
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0answers
288 views

Is there a name for this type of plot? (function on complex plane vs time shown in 3D)

I'm just looking for a name for this type of plot, which is time vs real part vs imaginary part shown as a space curve. Complex exponential: Used to explain chirplets: Complex Morlet wavelet ...
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1answer
359 views

Describe the set whose points satisfy the following relation.is region?

Describe the set whose points satisfy the following relation.is region? |z − 2| > |z − 3|. My atempt The open half-plane: Re z > 5/2; a region, My guess is that if this region takes all the ...
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2answers
1k views

Are irrational complex numbers possible?

I am asking because I was reading this and the mathematics is a little over my head. The title of the paper is Rational Approximations to Irrational Complex Number, and I didn't think that complex ...
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3answers
405 views

Expressing $e^z$ where $z=a+bi$ in polar form.

I am reading a passage of text that states: "We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
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3answers
193 views

What is the proper name for this number system?

The FractInt documentation makes mention of two number systems which extend the complex numbers: the "quaternions" and the "hypercomplex numbers". However, Wikipedia claims that "hypercomplex number" ...
4
votes
3answers
828 views

No extension to complex numbers?

Complex numbers are 2D. It is a commonly sited result that there is no 3D or 4D analogue of the complex numbers. I just want to be clear on exactly what this result says: It is impossible to ...
3
votes
1answer
340 views

Sum of every $k$th binomial coefficient.

It is widely known that $$\sum_{m=0}^{n} {n\choose m} = 2^n$$ and that $$\sum_{m=0}^{\lfloor\frac{n}{2}\rfloor}{n\choose 2m} = 2^{n-1}$$ Both results can be proven by exploting the nature of the roots ...
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2answers
106 views

complex equation

In Gaussian plane draw solution of equation |z-(1+2i)|=2 My solution: Wolfram solution: I don't understand, why my solution is not right. Could anyone help me, please?
5
votes
1answer
852 views

Why are imaginary numbers called imaginary numbers

Why do we call imaginary numbers "imaginary numbers"? As far as I can tell, there's nothing really imaginary about them. They exist. They're used all the time. What makes them so "imaginary"?
5
votes
1answer
160 views

Generalized addition function

I would like to have an example (or a proof that there does not exist) of a function on the complex numbers, which for lack of a better term I'll call generalized addition, such that $$x\oplus ...
2
votes
1answer
132 views

Root of Complex Number

I just want to clarify something here. Using elementary computation we can verify that for $x,y\in\mathbb{R}$ $$\sqrt{x+iy}=\pm\left(\sqrt{\frac{r+x}{2}}+i \sqrt{\frac{r-x}{2}}\right)$$ where ...
4
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1answer
2k views

Are complex determinants for matrices possible and if so, how can they be interpreted?

I've been asked to compute the determinant of a 3x3 matrix with complex entries. I have done so using the normal expansion along a row or column method that I would use were the entries real. My ...
0
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1answer
188 views

Eigenvalues of $A$ compared to $A^H$

How are the eigenvalues of $A^H$ related to the eigenvalues of $A$? Here $A^H$ is the conjugate transpose of $A$
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1answer
53 views

Can we proof $B=C^*$ given $|B-Ae^{-j\omega\delta}|=|C-Ae^{+j\omega\delta}|$

I want to ask for verification about whether this equation can be proven. If so, what is the best way to approach it? I tried this way... but I don't know how to continue on. ...
1
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1answer
974 views

Proof of Osborne's Rule

Osborne's rule is described here. Firstly, am I right that only signs of terms in the form $\sin^{4n+2} \theta$, $n \in \mathbb{Z^+}$ have their signs switched (i.e. terms like $\sin^4 \theta$ simply ...
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1answer
209 views

Complex Numbers and polar form

I am given the following information: $$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$ where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general. I was wondering how the following is ...
3
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7answers
2k views

Simple applications of complex numbers

I've been helping a high school student with his complex number homework (algebra, de Moivre's formula, etc.), and we came across the question of the "usefulness" of "imaginary" numbers - If there ...
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1answer
1k views

Complex number loci and min/max argument

The Question: I have some gaps in this chapter, and I would like some clarifications. What does arg(z) represent and what does $${\displaystyle \arg \left( z+2-2\, \sqrt{3}i \right) }$$ represent? ...
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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 ...
2
votes
2answers
1k views

A-stability of Heun method for ODEs

I'm trying to determine the stability region of the Heun method for ODEs by using the equation $y' = ky$, where $k$ is a complex number, based on the method described here. If the Heun method is: ...
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2answers
408 views

Determination of complex logarithm

Good day everyone. I was reading the more advanced lectures on complex analysis and encountered a lot of questions, concerning the determination of complex logarithm. As far I don't even understand ...
0
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1answer
149 views

Elementary Question about Roots of unity

What is the formula to find a specific root of unity? Also, what does a primitive root of unity mean? I know that $\zeta_5^5=1$ (5th root of unity), but how would I find $\zeta_5^2$? (the second ...
0
votes
1answer
141 views

How to solve this by galois theory?

please focus on the concept to solve this problem, because i can't handle to research on diffcult terminology.Thanks in advance. Find all real roots by galois theory and find all other root to this ...
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2answers
61 views

Is there a similar function in complex number system corresponding to logarithim in real number system?

i notice that there are $e^{i\theta}$ in math,so is there a similar function in complex number system corresponding to logarithim in real number system?
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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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6answers
827 views

Show that $z^6 + 5z^4 - z^3 + 3z$ has at least two real roots given that all roots are distinct.

Show that $z^6 + 5z^4 - z^3 + 3z$ has at least two real roots given that all roots are distinct. Also, show that $|3z - z^3 + 5z^4| < |z^6|$ when $|z| > 3$. I can see that 0 is a real ...
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2answers
10k views

Distance Between Any Two Points on a Unit Circle

As part of a larger investigation, I am required to be able to calculate the distance between any two points on a unit circle. I have tried to use cosine law but I can't determine any specific manner ...
3
votes
2answers
90 views

question about binomial expansion's coefficients

I am trying to show that if $$\left( 1+x\right) ^{n}=p_{0}+p_{1}x+p_{2}x^{2}+\ldots $$ and n being a positive integer, then $$p_{0}-p_{2}+p_{4}+\ldots = 2^{\frac {n} {2} }\cos \dfrac {n\pi } {4}$$ and ...
3
votes
1answer
126 views

inequality with modulus of complex number

Let $ \displaystyle{ z_1, z_2 \in \mathbb{C} }$ where $ z_1, z_2 \neq 0$ Prove that: $\displaystyle |z_1 +z_2| \geq \frac{1}{2} \left( |z_1|+|z_2| \right) \left|\frac{z_1}{|z_1|} + ...
5
votes
2answers
263 views

Primitive roots of unity

I am trying to show that, If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then $$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} ...
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1answer
93 views

Is there a forumla for number of primes preceding a natural number?

I am guessing there is no known analytical function which gives such a formula. This question came to mind while attempting a rather basic proof. I am trying to show that the number of primitive ...
0
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1answer
88 views

Please explain this formula

So I'm doing this course about image processing, which algorithm heavy course. Now there's a thing called Fourier transform. Here's few formulas that is used to explain the basics: $$C=R+jI$$ ...
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0answers
972 views

Physical meaning of Fourier transform of complex signal?

I understand what is meaning of Fourier transform over function that returns only real values — it can be thought of function taking time and returning real ...
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1answer
126 views

What mistake did I make finding $\arg (\sqrt{3}-i)$

What mistake did I make finding $\arg (\sqrt{3}-i)$? I figured it will be in the 4th quadrant and look like: So $\arg{z} = - \arctan{\frac{\sqrt{3}}{1}} = -\frac{\pi}{3}$ The right answer is ...