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

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5
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1answer
977 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
167 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
141 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
votes
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
votes
1answer
193 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 $A$
1
vote
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
vote
1answer
1k 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 ...
1
vote
1answer
217 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
votes
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 ...
1
vote
1answer
2k 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? ...
0
votes
1answer
61 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: ...
2
votes
2answers
441 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
votes
1answer
151 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
142 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 ...
1
vote
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?
0
votes
1answer
292 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. $\langle \phi| \hat A|\phi\rangle$ ...
1
vote
6answers
888 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 ...
1
vote
2answers
12k 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
91 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
131 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|} + ...
6
votes
2answers
276 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\} ...
0
votes
1answer
94 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
votes
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: $$C=R+jI$$ ...
2
votes
0answers
1k 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 ...
0
votes
1answer
164 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 ...
2
votes
4answers
140 views

Evaluate $(\frac{1}{-\sqrt{2}+\sqrt{2}i})^{2011}$

Evaluate $$(\frac{1}{-\sqrt{2}+\sqrt{2}i})^{2011}$$ So ... $$(\frac{1}{-\sqrt{2}+\sqrt{2}i})^{2011} = (-\sqrt{2}+\sqrt{2}i)^{-2011}$$ $$\theta=\pi - \arctan(\frac{\sqrt{2}}{\sqrt{2}}) = ...
1
vote
1answer
190 views

Trigonometric result concerning DeMoivre's formula.

Given this question is rather long to answer, and I'm losing hope it'll ever be, I just want an answer to this particular claim: Working on the unitary circle, let $x=1-\cos \theta$ and $t=1-\cos n ...
1
vote
3answers
387 views

Why is $|1+e^{i\phi}|=|2\cos(\phi/2)|$?

$$|1+e^{i\phi}|=|2\cos(\phi/2)|$$ Hey guys, just wondering why the above is true, I don't think I quite understand how argand diagrams work. Supposedly, using an argand diagram I should be able to ...
4
votes
2answers
1k views

Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$)

Most of us are aware of the famous "Basel Problem": $$\sum_{k=1}^\infty \frac{1}{k^2} = \frac{\pi^2}{6}$$ I remember reading an elegant proof for this using complex numbers to help find the value of ...
1
vote
1answer
115 views

A complex exponent equation

Given $0 \leq x < 1$, $0 < Re(\rho) < 1$. Then does this equation contain no solutions for x other than $x = \frac{1}{2}$? $$2^\rho + \frac{1}{2} = \frac{1}{(1-x)^\rho} + x$$ I am ...
2
votes
1answer
124 views

Rudin Question (Integration of Complex Functions) [pg.325]

I was reading Rudin and I stumbled upon a proof that I do not seem to understand. It is on page 325 of Baby Rudin $3^{rd}$ edition. In case you do not have a copy I shall write some background ...
1
vote
1answer
2k views

Cross product in complex vector spaces

When inner product is defined in complex vector space, conjugation is performed on one of the vectors. What about is the cross product of two complex 3D vectors? I suppose that one possible ...
2
votes
2answers
128 views

About the logarithm of the negative unit

$${e}^{iz} = \cos(z)+i\sin(z)$$ and $$e^{i\pi}=-1$$ But then $$\ln(-1)$$ can be infinite many numbers (positive and negative), as $z$ is the natural logarithm of that number and the solution to the ...
7
votes
2answers
331 views

How does one find $z\in \mathbb{C}$ such that $\sin z=100?$

I am self-studying Complex Analysis and I am suppose to find $z\in \mathbb{C}$ such that $\sin z=100.$ I know that $$\sin z=\sin x \cosh y+i\cos x\sinh y$$ So I must have $\sin x \cosh y=100.$ I ...
0
votes
1answer
61 views

Find complex $z$ such that $z$ has the largest possible real part, and satisfies: $z^7 = -18-18i$

Find complex $z$ such that $z$ has the largest possible real part, and satisfies the equation: $z^7 = -18 -18i$ So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ ...
2
votes
4answers
447 views

For complex $z$, find all solutions to: $(z - 6 + i)^3 = -27$

For complex $z$, find all solutions to: $(z - 6 + i)^3 = -27$ I reasoned that for this to be true, $z - 6 + i$ must be $= -3$ $\therefore z - 3 + i = 0$ $z = 3 - i$ However, there are two more ...
4
votes
6answers
3k views

Plot $|z - i| + |z + i| = 16$ on the complex plane

Plot $|z - i| + |z + i| = 16$ on the complex plane Conceptually I can see what is going on. I am going to be drawing the set of points who's combine distance between $i$ and $-i = 16$, which will ...
0
votes
3answers
232 views

Write $\cos(9x)$ in terms of powers of $\cos(x)$ [duplicate]

Possible Duplicate: How to expand $\cos nx$ with $\cos x$? Write $\cos(9x)$ in terms of powers of $\cos(x)$ I realize I could solve this by using De Moivre's and binomial expansion: ...
1
vote
0answers
90 views

Recover complex number from several noised components.

I know these 12 values, which are derivative from the same unknown complex variable z: ...
2
votes
2answers
61 views

Analyze $x^A(1 - x)^B$ divided by $(1 + x^2)$ with remainder $ax + b$

Assume $A, B \in \mathbb{Z}^+$. If $x^A(1 - x)^B$ is divided by $(1 + x^2)$, the remainder is $ax + b$, show that $a = (\sqrt{2})^B \sin\frac{(2A - B)\pi}{4}$ and $b = (\sqrt{2})^B ...
-2
votes
2answers
82 views

Another question on radicals [closed]

I'm back, and again with radicals and complex numbers; I have the follow multiplication: $$2\sqrt{-2} \cdot 3\sqrt{-3} = $$ Working: $$2\sqrt{2i} \cdot 3\sqrt{3i} = $$ And now, the doubt: ...
12
votes
5answers
864 views

What's bad about calling $i$ “the square root of -1”?

I vaguely recall a teacher telling me that he dislikes introducing the imaginary unit $i$ as "the square root of $-1$", but I can't remember why. Is there a lack of rigour in the statement, or is it a ...
1
vote
2answers
3k views

How to find the Euclidean norm of a complex number?

How to find the Euclidean norm of a complex number, like $10+i$ or $2-i$? Explain with clear, easy detail.
4
votes
2answers
271 views

How do I prove the following statement about a summation of a series?

I have not been able to completely solve this problem and it's driving me crazy. Could you please help. The question is to show that, $$\sum_{n=1}^N \frac{\sin n\theta}{2^n} ...
3
votes
1answer
59 views

Compute $\sum_{i=1}^{2n} \frac{x^{2i}}{x^i-1}$ where $\{ x \in \mathbb{C}$ | $x^{2n+1} = 1, x \neq 1\}$

$\{ x \in \mathbb{C}$ | $x^{2n+1} = 1$ , $x \neq 1\}$ Compute $\displaystyle{\sum_{i=1}^{2n} \frac{x^{2i}}{x^i-1}}$
1
vote
2answers
98 views

Exponential Equation with mistaken result

I'm on my math book studying exponential equations, and I got stuck on this Problem: What is sum of the roots of the equation: $$\frac{16^x + 64}{5} = 4^x + 4$$ I decided to changed: $4^x$ by $m$, ...
6
votes
1answer
417 views

How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
0
votes
5answers
380 views

Simple doubt about complex numbers

The question itself is simple, but I'm weak in math, and I'm training a lot every day to be the best I can, so, working on complex numbers, I got stuck on a simple multiplication: $\sqrt{3i} * 2i$ ...
3
votes
2answers
824 views

The Fundamental Theorem of Algebra and Complex Numbers

We had a quiz recently in a linear algebra course, and one of the true/false question states that The Fundamental Theorem of Algebra asserts that addition, subtraction, multiplication and division ...