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

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1answer
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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 ...
2
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4answers
38 views

Calculating real and imaginary part of a complex number

Consider the complex numbers $a = \frac{(1+i)^5}{(1-i)^3}$ and $b = e^{3-\pi i}$. How do I calculate the real and imaginary part of these numbers? What is the general approach to calculate these ...
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1answer
16 views

Converting complex number raised to a power to polar form

How would u convert (1+i)^n to polar form? I've heard about de Morgans law but I don't know how to apply it here
5
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1answer
293 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 ...
4
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1answer
138 views

Laplace transform of and impulse sampled function using “frequency” convolution

This is a long question, but assume we have this: The book uses the frequency convolution theorem to solve this problem. To solve the integral, it uses a contour + residue theorem to solve it. The ...
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5answers
82 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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1answer
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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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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 ...
3
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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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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 ...
3
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4answers
70 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
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 ...
0
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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 ...
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2answers
41 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 $$ ...
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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 ...
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
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 ...
1
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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} ...
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0answers
46 views

Is derivation of Feigenbaum constant possible through Mandelbrot set?

this is Mandelbrot set: $z_{n+1}=z_n^2+C$ Is derivation of Feigenbaum constant possible through Mandelbrot set? $$\lim_{n\to\infty}\frac{z_{n+2}-z_{n+1}}{z_{n+1}-z_{n}}=\delta$$
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1answer
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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
23 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
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
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
14 views

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
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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
47 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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6answers
9k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
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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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2answers
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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
41 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 ...
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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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7answers
2k views

What does it mean to divide a complex number by another complex number?

Suppose I have: $w=2+3i$ and $x=1+2i$. What does it really mean to divide $w$ by $x$? EDIT: I am sorry that I did not tell my question precisely. (What you all told me turned out to be already known ...
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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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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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1answer
20 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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0answers
24 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
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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) ...
2
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2answers
30 views

Problem converting to polar form in proof

I wonder if anyone has an idea about how to write $$ \prod_{\substack{j=0\\ j\neq k}}^{n-1} ( e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi ij}{n}})=n,\qquad k=0,1,...,n-1,\; j=0,1,....n-1$$ in a "general" ...
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
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?
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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
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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
31 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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2answers
38 views

Using the formulas de Moivre to deduce trigonometric identities.

Yesterday I made a test of complex variables, and this contained a question (in which I could not solve) that asked to use the de Moivre formulas to deduce the following trigonometric identities: ...
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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 ...