Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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Complex numbers

3 questions, not sure how to do them. Let $z$ and $w$ be complex numbers such that $w=\frac{1}{1-z}$ and $|z|^2=1$. Find the real part of w. If $z=e^{i\theta}$ prove that ...
2
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2answers
122 views

Evaluation of complex real numbers

The much anticipated math.SE community blog will $\tiny\mathrm{hopefully}$ contain a contribution from Alex Becker with the topic The Complex Real Roots of $x^3-3x+1$, which I'm really looking forward ...
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3answers
90 views

Prove a function is not continous

How can I (formally correct) prove that $\;f: \mathbb{C} \rightarrow \mathbb{C}$ = $ \left\{ \begin{array}{cl} 0 & z = 0 \\ e^{-\frac{1}{z^2}} & z \neq 0 \end{array} \right.$ is not ...
4
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2answers
77 views

About Gaussian Integers .

If a+bi is a Gaussian integer with (a,b)=1 call it 'viewable' ( as a line from a+bi to 0 can be drawn intersecting no other gaussian integers). Are there any gaussian composites that are viewable?
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1answer
56 views

$\sin\left(1+\frac{1}{z-1}\right)$ expanded in powers of $z-1$

The whole problem: Obtain the following Laurent expansions. State the first four nonzero terms. State explicitly the $n$th term in the series, and state the largest possible annular domain in which ...
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1answer
185 views

Complex Exponential False “Proof” That All Integers Are $0$

The following false "proof" is attributed to Thomas Clausen in 1827, and was stated on page 79 of Nahin's An Imaginary Tale. $e^{i2\pi n}=1$ for all integers $n$. So \begin{align*} ee^{i2\pi ...
6
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3answers
114 views

What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
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3answers
35 views

Complex number with real part as 0

it is kinda of awkward, but is Equation: 0+3i=0? Or it simply means that it is imaginary number?
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2answers
37 views

To find the value of complex number

If $z$ and $w$ are two non zero complex numbers such that $|zw| =1$ and $\arg z - \arg w = \pi/2$ then conjugate of $(zw)$ =?
2
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2answers
78 views

Is this a valid proof for eulers formula?

I am wondering whether this proof is a valid proof of Eulers formula: $e^{ix}=i\sin(x)+\cos(x)$ $$\frac{d}{dx}e^{ix} = i(e^{ix})$$ $$\frac{d}{dx}(i\sin(x)+\cos(x)) = i\cos(x)-\sin(x) = ...
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1answer
48 views

Why take the complex conjugate

If I have the equation: $$\frac{250+0.915j}{350+0.915j}$$ Why do I have to take the complex conjugate to rid of the complex part on the bottom? Why can't I just times by: $$ ...
2
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1answer
84 views

Prove that $\sum_{n=0}^{\infty}e^{in\theta}$ is bounded

For my homework class, we need to prove that a certain series converges, for which it is useful to use that this series is bounded ($\theta \in (0,2\pi)$): $$\sum_{n=0}^{\infty}e^{in\theta}$$ I ...
2
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2answers
47 views

Finding matching roots

If ${4 + \sqrt{2}}$ is one root of a quadratic equation given by ${x^2 - Px + Q =0}$ where P and Q are rational numbers then find the missing root. The answer is ${4 - \sqrt{2}}$. And I'm a bit ...
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1answer
37 views

Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?

I wonder about the second step of the proof shown below (d) in the picture attached. Why is $\langle y,x\rangle +\langle x,y\rangle=2\Re \langle x,y\rangle$?
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3answers
53 views

How to calculate $\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}$ with $z\in\mathbb{C}$ and $|z|\ne 1$?

As stated in the title: How does one calculate $$\displaystyle\lim_{n\to\infty}\sqrt[n]{|1-z^n|}$$ with $z\in\mathbb{C}$ and $|z|\ne 1$?
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0answers
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Complex integral and parametrization of a circle

I am trying to compute the following integral of $$\int \frac{1}{z^3+3} dz$$ over a circle of radius $2$, centerd at $(2,0)$. Thus I am trying to compute the residue and have found that the function ...
2
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2answers
31 views

Basic complex variable proposition

I have to prove the following property, but no idea how to start... I was told to solve it with polar coordinates, but I still don't know how. Let $\tau$ be a complex number with positive imaginary ...
4
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1answer
37 views

Set of Points in the Complex Plane

I'm having trouble describing the set: $\{z\in\mathbb{C}:|z-a|=r|z-b|\}$ where $r$ is a positive real number and $a,b$ are fixed complex numbers. I worked out the algebra and it seems to be a (real) ...
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3answers
76 views

Is the modulus of i^n 1 for all n?

As the tile says, is $\left | i^{n} \right | = 1$ for all real values of n?
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3answers
2k views

Is this a valid method of finding magnitude of complex fraction

If I have a complex fraction $\dfrac{a+bi}{c+di}$ and I want the magnitude, then will it be $\left|\dfrac{a+bi}{c+di}\right|=\dfrac{|a+bi|}{|c+di|}$? Scratch that ... I just found the answer on ...
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4answers
150 views

Values for $(1+i)^{2/3}$

This question might seem easier than I'm making it seem. But how many values are there for $(1+i)^{2/3}$? Do I let $z=(1+i)^{2/3}$ so that $z^3=2i$? I'm asked to write each in polar coordinates and in ...
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1answer
21 views

Is $\|x_1\|^2 + 2\|x_2\|^2 > - 2\Re(ix_1\overline{x_2})$ for complex numbers $x_1,x_2$

This is the last piece I need for a proof for a homework problem. Could someone explain whether or not this inequality must hold?
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2answers
35 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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1answer
75 views

Generalised Pythagorean Theorem?

$|a+b|^2=|a|^2+|b|^2+2 Re(\overline ab)$ Can anyone explain this equality to me? How it is derived?
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31 views

Understanding basic complex number rules

Many definitions for complex number say $Re(z) = \frac{1}{2}(z + \bar{z})$ $Im(z) = \frac{1}{2i}(z - \bar{z})$ $|z| = \sqrt{z\cdot\bar{z}}$ I do understand 1. as I can visualize it (the addition ...
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2answers
237 views

Calculating absolute value and argument of a complex number

I want to calculate the absolute value and argument of the complex number $a = \left(\sqrt{3} - i\right)^{-2}$. In order to calculate these two values I tried to reform the number into the form $z = ...
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4answers
3k 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
148 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
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3answers
467 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
180 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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2answers
44 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
156 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...
0
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1answer
78 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
72 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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1answer
75 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
117 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
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2answers
104 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
70 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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1answer
157 views

How to deal with $\bar{x}$ 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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4answers
66 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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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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78 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
74 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
57 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
47 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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3answers
58 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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1answer
75 views

From the equation $|z-a|=|z-b|$ in the complex plane, obtain the slope-intercept equation of the line

I want, by the use of the equation of the line in complex plane, to find the slope and x intercept in x-y plane. Attempt: $$|z-a|=|z-b|$$ $$y=mx+h$$ $$m=m(ax,ay,bx,by) \quad h=h(ax,ay,bx,by)$$ ...
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1answer
40 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
48 views

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) ...