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

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1answer
25 views

Group Theory proving [on hold]

can someone help me with this question? 1) Given a natural number n≥1, let $G_n$ be the set of complex n-th roots of $1$, i.e. $G_{n} = \{z \in \mathbb{C} :z^n = 1\}$ Prove that $G_n$ is a group ...
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3answers
53 views

Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
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5answers
79 views

How to teach newbie multiply of complex number

I want to teach a newbie the arithmetic law of complex numbers. the law of add is acceptable psychological. but multiply is not. for example, assume $$z = a+bi, w = c+di$$ He (She) may ask me: why ...
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0answers
16 views

Standard deviation on absolute square of complex average

I am calculating a quantity $q=|c|^2$ where I obtain $c\in\boldsymbol C$ as an average of a collection of estimates with errors: $\langle c\rangle=\sum_{j=1}^nc_j$, and the question is what error to ...
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0answers
17 views

Are there any applications of sedenions?

I've been interested in hypercomplex number systems for a while as fascinating little toys, all the way up to octonions; But the octonions look like they're the end of the line of interesting math as ...
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2answers
29 views

Find real domain of a function results in $x \geq i$

I have an equation of the form $$f(x) = \sqrt{x^3 + x}$$ for which one needs to define the maximal domain, and image and domain are part of $\mathbb{R}$ (real numbers). $$x^3 + x \geq 0 \implies ...
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0answers
15 views

Transcendental functions for $c\in\mathbb{C}$

Recently I have been writing a Julia set renderer, and I have now gotten to the polynomial stage where it can render any set of the form $z=P(z)+c$ where P is an arbitrary polynomial in z. Wikipedia ...
3
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4answers
68 views

Solve $z^2 - iz = |z - i|$

I have the equation: $z^2 - iz = |z - i|$ The solutions are $i$, $-\sqrt3/2 + i/2$, $\sqrt3/2 + i/2$ Can someone please walk me through or give me a hint...
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0answers
6 views

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity $\infty$?

How the extension of complex plane with complex infinity $\tilde{\infty}$ coexists with extension of real line with positive infinity? Are there any paradoxes arizing? What are the rules when the ...
0
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0answers
17 views

Derivatives, Cauchy-Riemann Equations [on hold]

Given the function, $w=z^4$ and I want to find the following solutions for this equation, Find real functions u and v such that w=u+iv Show that Cauchy-Riemann equation holds at all points in the ...
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2answers
40 views

Uniform convergence of the series

Test the uniform convergence of the series $$ \sum_{n=1}^\infty \frac{1}{z^2 - n^2 \pi^2}$$ $$ \forall z \not= \pm n\pi,\;\; where n \in\mathbb N$$ Can I find $M_n$ such that $$ ...
2
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2answers
32 views

Express each function in the form $u(x,y) + iv (x,y)$

I was doing some homework with complex numbers and I'm stuck with these two, I hope that someone can solve these and clear it up for me. Thank you. ln(1+z) z/(3+z) Samples,
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0answers
16 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
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1answer
24 views

Finding the locus represented by complex variable equations?

I'm trying to solve these two problems related to complex number but hardly found a solution. I hope that someone can solve these and clear it up for me. Thank you. |z+2|=2|z-1| |z+5|-|z-5|=6
3
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2answers
50 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
0
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1answer
47 views

Number of zeros of $ z^7+4z^4+z^3+1$

How many zeros does $z^7+4z^4+z^3+1$ have in each of the regions |z|<1 and |z|<2? I know I should use Rouche's Theorem but I can't find a $|f(z)| > |p(z)-f(z)|$ and $f(z)$ have equal number ...
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0answers
20 views

How to deal with x* when solving complex-variable linear equation(s) of x?

The theory of linear algebra can be directly applied to linear equation(s) of complex variables with the form \begin{equation} \sum_i a_i x_i=c\ldots\ldots(1) \end{equation} with $a_i,c\in ...
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1answer
28 views

What are $a$ and $b$ when the zeropoints of $f(z)=(a+bi)z+2-i=0$ is at $1-i$?

$f(z)=(a+bi)z+2-i$. What are the values of a and b when $1-i$ is the zeropoint of f? $f(z)=(a+bi)z+2-i=0$ $(a+bi)(1-i)+2-i=0$ $a+bi-ai-bi^2+2-i = 0$ $(a+b+2)+(-a+b-1)i=0$ I don't know what the ...
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0answers
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Meaning or use for complex slopes

In John Derbyshire's "Unknown Quantity" he mentions that lines with a complex slope can be perpendicular to themselves. I can easily prove this to myself using the definition of perpendicular slopes ...
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4answers
51 views

Proof that the coefficients of a polynomial are real

How does one prove that all the coefficients of this polynomial: $$(x+i)^{10}+(x-i)^{10}$$ are real numbers, without using Newton's Binomial Theorem?
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1answer
21 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
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1answer
29 views

Matrix representation of complex numbers in exponential form

Do there exist matrices M and P for this equation? Or perhaps M and P dont need to be matrices? I saw this and this question after googling which made me wonder about whether the exponential form of ...
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4answers
46 views

Solving $|z|i+2z=\sqrt{3}$

How one can solve the following complex equation, where $z$ is complex number. $$|z|i+2z=\sqrt{3}$$ Thank you.
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2answers
38 views

the geometric explain of $t = x-\frac{a}{3}$ in the simplify of cubic equation $x^3+ax^2+bx+c=0$

Assume $$f(x) = x^3+ax^2+bx+c$$ we have $$f''(x)=2a+6x$$. we get $x = -\frac{a}{3}$ Magically, If we take the transformation: $$t = x -\left(-\frac{a}{3}\right)$$. we can transform the above ...
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1answer
40 views

Value of $i^2$ in complex numbers [duplicate]

Please solve this doubt : we know that $\sqrt{a}\sqrt{b}=\sqrt{ab}$ and $i^2 = -1$. But $i= \sqrt{-1}$ which implies that $i^2 = i \cdot i = \sqrt{-1}\sqrt{-1} = \sqrt{1} = 1$ that is $i^2 = 1$. So ...
0
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3answers
49 views

Algebraic Equation?

$$Ve^{i\theta} = We^{i\phi}$$ where, $V$ and $W$ are some real constants. From this my book concludes: $\theta = \phi$. How does it conclude this? I don't see why its valid to just equate the ...
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0answers
39 views

Complex number from its roots

I need help to figure out if I solved this exercise correctly because I don't have the exercise outcome: Fifth roots of a complex number $ \ z \ $ have been calculated: find $ \ z \ $ knowing that the ...
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0answers
30 views

Locus to two complex number given. least value of their difference needs to be found.

there are two complex numbers defined by locus arg(w-2)=3/4 pi and |w+i|=1 the blue line shows |w-z| we need to find the least value of |w-z| how can we do it. i thought about z being √2/2 - √2/2 i ...
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0answers
23 views

Complex Number Plane Physics Math Problem

Show that if the line through the origin and the point z is rotated 90°about the origin, it becomes the line through the origin and the point iz. Use this idea in the following problem: Let ...
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1answer
32 views

Is it possible to find a formula for $d$ in terms of $a$, $b$, and $c$?

If $a$, $b$, $c$, and $d$ are complex numbers on the unit circle, and $\overline{ab}\perp\overline{cd}$, is it possible to find a formula for $d$ in terms of $a$, $b$, and $c$?
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1answer
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Sinusoids closed under addition, Euler's Formula

Real sinusoids with the same frequency are closed under addition. If $$f(\omega) = A_1 \cos(\omega + \phi_1) + A_2 \cos(\omega + \phi_2)$$ Then there is some $A_3$ and $\phi_3$ so that: $$f(\omega) ...
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2answers
65 views

Complex numbers confused!!

If you give me a complex number say $z=2+3i$, then I can easily find $\text{Im}(z)=3$ and $\text{Re}(z)=2$ but when this polar coordinates stuff came, I lost my head! So say ...
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0answers
29 views

Root of unity paradox

Suppose $w$ is a cube root of unity. Then we know that $w^3 =1$. now suppose we want the value of $w^4$. $w^4 = (w^3)^{4/3} = 1^{4/3} = 1$ which is obviously false Why does this happen?
2
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1answer
30 views

How to show $\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$

How to show $$\sin(x+iy)=\sin(x) \cosh(y) + i\cos(x) \sinh(y)$$ I begin with $$\sin(x+iy) = \frac{e^{x+iy}-e^{-x-iy}}{2i} = \frac{e^xe^{iy}-e^{-x}e^{-iy}}{2i}$$ $$ = ...
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1answer
47 views

Finding argument of a complex number

How do you evaluate the following $$\text{Arg}\{\sin\frac{8\pi}{5} + i(1 + \cos\frac{8\pi}{5})\}$$
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1answer
25 views

Plot of a domain in the complex plane

I am trying to plot the following domain in the complex plane: $\lbrace x\in\mathbb{C}|\: |x^{2}-1|<r\rbrace$ for some $r>1$. I know that in general to take a square root of a complex number ...
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1answer
12 views

Residues more than one singularity at 0

Having trouble calculating the residue at 0 for this integral within the unit circle I understand that its a pole of order 3 because both the z^2 and the sinz have singularities at 0. Is there an ...
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0answers
24 views

Divisibility by $z-z_0$ if $z_0\in \mathbb{C}$ [duplicate]

I have a problem I'm working on, and I'm just not getting it. Suppose that $z_0\in\mathbb{C}$ is fixed. Show that if $P(z)=c(z^k-z_0^k)$, then there exists a polynomial $Q(z)$ such that ...
0
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1answer
55 views

Complex number, strangely written

Find all the complex solutions of the equation: $$\frac{z^3}{i} = 1$$ I mean is this the same thing as $$z^3 = i$$? Because I don't understand why my teacher would put it like that on a test. At ...
3
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1answer
70 views

What are these numbers?? (floor(a)=0)

I was just so I decided to go and look up the roots of floor(a) in WolframAlpha where a is any number, real or complex, and of course the interval [0,1) showed up as an answer but I also got these ...
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0answers
33 views

Test the uniform convergence of the series in indicated region

Test the uniform convergence of the series I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $ by using Abel's Theorem This is the question : Test the ...
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1answer
15 views

Complex Eigenvalue Periodic

This is probably a simple question, but I'm having trouble finding a clear answer. Let's say we have a system, with the complex eigenvalue: $$\lambda = \alpha + i\beta$$ I know that $\alpha < 0$ ...
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2answers
30 views

Finding numbers $a$ and $b$ for a complex number

Problem. Given a complex number $$z=2-2i$$ Find numbers $a$ and $b$ such that $$a+ib = \frac{1}{z}$$ I tried multiplying both sides by $z$ and got $$(a+ib)(2-2i)$$ $$= 2a-2ai+2bi-2bi^2$$ ...
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2answers
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Why does $| a_n + \frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} | \ge | a_n | - |\frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} |$ hold?

Why does $$| a_n + \frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} | \ge | a_n | - |\frac {a_{n-1}} {z} + \ldots + \frac {a_{0}} {z^n} |$$ hold by the trinagle inequality for $z, a_i \in \mathbb C$ ...
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1answer
39 views

arg(z) vs. Arg(z)

What is the difference between the arg(z) and the Arg(z), where z is a complex number of the form a+bi for example z = -2 - 2i the angle from the positive ...
0
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1answer
19 views

Functional Equation and totally multiplicative functions

Find all $f:\mathbb{C}\rightarrow\mathbb{C}$ s.t. $\forall a, b\in \mathbb{C}, f(ab) = f(a)f(b)$. I could deduce a lot of things about what happens at the roots of unity, and 0, but I can't find out ...
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3answers
29 views

Finding an expression for the complex number Z^-1

So I want to find out an expression to express: $$z^{-1}$$ I know the answer is: $$z^{-1} = \frac{x-iy}{x^2+y^2}$$ But how would I go about proving this/the steps to this?
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1answer
58 views

Complex number with 3 dimensions [duplicate]

I was looking back on complex analysis and asked myself: ''Why is there no complex number in 3 dimensions ?''. To place this question let me define with what I mean with 3 dimensions in the following. ...
0
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1answer
69 views

Why isn't $i$ affected by powers?

When finding roots of complex functions we can write for example: $$z=2-2i$$ Let's find complex numbers $w$ such that $$w^4 = 2-2i$$ $$\large z = \sqrt{8} e^{ \frac{- \pi }{4} i}$$ This reads: ...
1
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1answer
23 views

Complex numbers

In an Argand diagram, the vertices on an equilateral triangle lie on a circle with center at the origin. One of the vertices represents the complex numbers 4 + 2i. Find the complex numbers that ...