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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1
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1answer
48 views

Solve complex equation $5|z|^3+2+3 (\bar z) ^6=0$

I'm stuck in trying to solve this complex equation $$ 5|z|^3+2+3 (\bar z)^6=0$$ where $\bar z$ is the complex conjugate. Here's my reasoning: using $z= \rho e^{i \theta}$ I would write $$ ...
-2
votes
1answer
69 views

Will this punch a hole in the field of complex number? [closed]

According to this, complex number is algebraically closed, i.e. every polynomial has complex root. What if we allow other type of equations? I ask this question because equations seemingly can extent ...
1
vote
1answer
40 views

inequality on the unit disk

$n$ points $z_1,z_2,\cdots,z_n$ in the unit open disk are given. Prove or disprove that there exists $z$ in the unit circle such that $\prod_{i=1}^n |z-z_i|^i \ge 1$. I think it can be solved by ...
7
votes
1answer
176 views

Complex Numbers $\stackrel{?}{=} \mathbb{R}^ 2$

Suppose we have a vector field over real numbers $\mathbb R^2$. In additon to vector field proporties define inner product $(x,y) = x_1\cdot y_1 + x_2\cdot y_2$, where $x_1,x_2,y_1,y_2$ are real ...
1
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3answers
41 views

Question about complex numbers [duplicate]

Proof that if $z = 1$, then $|z-w| = |1- \overline{w}z|$, $\forall w \in$ $\mathbb{C}$ My attempt below: $(z-w)\overline{(z-w)} = |z-w|^2$ $(z-w)\overline{(z-w)} = (z-w)(\overline{z}-\overline{w}) = ...
3
votes
0answers
91 views

A question about the proof of $(z_1z_2)^a=z_1^az_2^a$

For $z_1,z_2\in \mathbb C$ if $\Im(z_1z_2)>0$ and $\Im(z_2)\ge 0$ prove that $(z_1z_2)^a=z_1^az_2^a$ , for $a$ is any real. I proved it like this: $z_1^az_2^a=\exp(a\log z_1)\exp(a\log ...
1
vote
1answer
23 views

Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$.

Let $f$ be analytic in an open set $U \subseteq \Bbb C$ and let $K \subseteq U$ be compact. Show that there exists a constant C depending on $U$ and $K$ such that $|f(z)| \leq C(\int_{U} |f|^2)^{1/2}$ ...
1
vote
0answers
47 views

Practical application of Gauss-Lucas theorem

Let $z_1,z_2,z_3 \in \mathbb C$ pairwise distinct be the affix of points $A, B$ and $C$. Let $P(x)=(x-z_1)(x-z_2)(x-z_3)$. Let $z_4$ and $z_5$ be the roots of $P'$ (with the possibilty that ...
0
votes
1answer
28 views

What region of the complex plane does $\left|{z-1+i}\right|+\left|{z+1-i}\right|=6$ fill?

What region of the complex plane does $\left|{z-1+i}\right|+\left|{z+1-i}\right|=6$ fill? I'm having a tough time figuring what region this fills up. Maybe its easy, but for some reason I cant think ...
0
votes
3answers
57 views

Is $\log(-1)$ equal to $-\log(-1)$ [duplicate]

I thought it should be because if the logarithmic identities hold then, $$-\log(-1)=\log(-1^{-1})=\log(-1)$$ But $\log(-1)=i*\pi$ and $-\log(-1)=-i*\pi$
4
votes
1answer
33 views

Perfect number in gaussian integers

We have complete description about irreducibles in the ring Z[i],of gaussian integers. Now I was trying to define suitably the notion of "perfect number" in Z[i]. But the problem is unique ...
0
votes
6answers
60 views

Find the real and imaginary parts of z

How can I start to solve this kind of equation ? Kind of stuck on getting the right answer too. $$\frac{1}{z}=\frac{2}{2+j3}+\frac{1}{3-j2}$$ Thanks for helping in advance!
1
vote
1answer
41 views

computing the cubed root of a complex number…

I do know how to calculate the cubed root of a complex number....like if I'm given that $x^3=p$, where $p$ is a complex number, then $$x= r^{1/3}\left(\cos\left(\frac{2k\pi+m}{3}\right) + i\sin ...
1
vote
4answers
80 views

Show that either $1+\alpha+\alpha^2+…+\alpha^{p-1}=0$ or $1+\alpha+\alpha^2+…+\alpha^{q-1}=0$,but not both together.

Let a complex number $\alpha,\alpha\neq1$,be a root of the equation $z^{p+q}-z^p-z^q+1=0$,where $p$ and $q$ are distinct primes.Show that either $1+\alpha+\alpha^2+.....+\alpha^{p-1}=0$ or ...
0
votes
2answers
38 views

Find the image of a triangle under the mapping $w=1/z$ [closed]

The triangle is $\{z=x+iy: x \geq 0, y \geq 0, x+y\leq1\}$. Could anyone help me with this? I also wonder how to solve this kind of problem in general. Specifically, I want to how the hypotenuse gets ...
2
votes
3answers
44 views

Formula for raising a complex number to a power

Is there an existing formula to raise a complex number to a power? That is, I want to compute $(1 + i)^N$. I basically want to write a function like so: ...
-1
votes
3answers
44 views

Clarification regarding a question

In the question in the link is it compulsory that $A+B+C=\pi$ ? If sin A +sin B+sin C = cos A+cos B+cos C=0 prove that sin 2A+sin 2B+sin 2C =cos 2A+cos 2B+cos 2C
1
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4answers
92 views

If $\sin A +\sin B+\sin C = \cos A+\cos B+\cos C=0$ prove that $\sin 2A+\sin 2B+\sin 2C =\cos 2A+\cos 2B+\cos 2C$ [duplicate]

If $\sin A +\sin B+\sin C = \cos A+\cos B+\cos C=0$ then how to prove that $\sin 2A+\sin 2B+\sin 2C =\cos 2A+\cos 2B+\cos 2C $? I tried using complex numbers but it's not working. Please help !
0
votes
3answers
67 views

Complex numbers - Why am I missing solutions?

Please explain to me why: $$ Z^2 = \bar{Z} $$ Has a 3rd and 4th solutions: $1, 0$ ? I found the first two complex numbers, but why also $1$ and $0$ ? I replaced $Z^2$ with $(x+yi)^2$ and $\bar{Z}$ ...
2
votes
2answers
39 views

Is “the nth root of x” well-defined without further qualification?

(No, I'm not asking if $\sqrt{-1} = +i$ or if $\sqrt{-1} = -i$. Yes, I know $+i$ is the principal square root.) Consider the cube root of -8. If asked to evaluate it, I would say -2, and I think we ...
0
votes
0answers
26 views

Question about a certain type of polynomials

Let $p:\mathbb{C}\rightarrow\mathbb{C}$ be a complex polynomial of degree $n\in\mathbb{N}$ and that $\text{Re }p(z)\geq0$ when $z\in\mathbb{R}$. Moreover impose that $p(z)$ has a point ...
2
votes
5answers
89 views

How to factor $4x^2 + 2x + 1$?

I want to know how to factor $4x^2 + 2x + 1$? I found the roots using quadratic equation and got $-1 + \sqrt{-3}$ and $-1 - \sqrt{-3}$, so I thought the factors would be $(x - (-1 + \sqrt{-3}))$ and ...
0
votes
2answers
39 views

Is there a word for the value of a complex number multiplied by its conjugate?

For a complex number $w$, or $a+bi$, is there a specific term for the value $w\overline{w}$, or $a^2+b^2$?
2
votes
1answer
14 views

How to deal with complex eigenvalues when computing fractional anisotropy.

I am attempting to measure the laminar flow, or rather how non-laminar a velocity field is. In order to do this I am looking at Fractional Anisotropy. The FA is calculated from the eigen values of ...
2
votes
2answers
58 views

a matrix of rank $r$ satisfies a polynomial of degree $r+1$.

Let $M$ be an $n\times n$ matrix with coefficients in $\mathbb C$. Suppose $M$ has rank $r$ with $r<n$. Prove there is a polynomial $P(x)$ with degree $r+1$ and coefficients in $\mathbb C$ such ...
1
vote
1answer
60 views

Extending the complex numbers by the solution of $|x| = -1$

I don't think I've ever encountered a situation where I've wanted to solve equations of the form $|x| = -1$, but you often hear that mathematics should be explored for the sake of mathematics. I'm ...
5
votes
3answers
37 views

Solve the binomial equation

Solve the binomial equation $$z^4 = -8$$ Below is the steps i have done 1: I have taken |-8| that is 8 and then done 8^(1/4) which is 2^(1/4). 2: Since $z=r(cos\alpha+isin\alpha)$ leads me to ...
1
vote
2answers
29 views

Exponential functions with negative base

Consider the function $f(x) = (-2)^x$, $x$ belongs to irrationals. For which $x$ does $f(x)$ belong to the reals.
0
votes
2answers
29 views

Minimizing this complex expression

I am working through Ahlfors' Complex Analysis book. I have come to the section in Chapter 1 on inequalities. Among the exercises in this section is this: Given complex numbers $a$ and $b$, choose ...
0
votes
0answers
15 views

What are the different forms in which the complex equation of a circle can be represented except |z-a|=r?

What are the different forms in which the complex equation of a circle can be represented except |z-a|=r?How to convert them to |z-a|=r form? P.S:I am not being able to find a good link/site ...
1
vote
2answers
56 views

Product of complex solutions via factorisation

I'm wondering if someone could help me out. I am asked to solve the equation: $z^6 =−1$ in part (a) of a question. I have done this and so I now have a set of solutions: $z_0,z_1,z_2,z_3,z_4.$ ...
1
vote
1answer
34 views

How to plot $arg(\frac{z-a}{z-b})=\theta$ using intuition? How to find its Cartesian equation?

How to plot $arg(\frac{z-a}{z-b})=\theta$ using intuition? How to find its Cartesian equation? z is a complex number.
4
votes
5answers
71 views

Solve a complex equation

Solve the following equation $$(4-3i)z^2-25z+31-17i= 0 $$ Dividing by 4-3i gives me $$z^2 \frac{-100z-75zi + 124 + 93i -68i -51i^2}{25}$$ which goes to $$z^2 -4z-3zi + 7+i$$ then i collect the ...
1
vote
4answers
381 views

Solve the complex equation

The equation is $$z^2 -4z +4+ 2i = 0$$ I know that i am supposed to use $$(a+bi)^2 = a^2 + 2abi + bi^2$$ to solve the equation but i am stuck on how to expand the equation. Can you help out with ...
2
votes
0answers
50 views

Quaternion rotation intuition

Say the quaternions real and imaginary part are written as $(q_1, \vec q)$. One useful multiplication property is $qr=(q_1r_1 - \langle\vec q, \vec r\rangle, q_1\vec r + r_1\vec q + \vec q \times \vec ...
3
votes
3answers
80 views

Simple confusion in complex analysis

I've been learning Complex Analysis from George Cain's website: https://people.math.gatech.edu/~cain/winter99/complex.html In chapter 3, Elementary Functions, it claims that the complex logarithm ...
1
vote
3answers
36 views

Proving $|z_1z_2|=|z_1||z_2|$ using exponential form of a Complex Number

Problem: Prove $$|z_1z_2|=|z_1||z_2|$$ where $z_1,z_2$ are Complex Numbers. I tried to solve this using the exponential form of a Complex Number. Assuming $z_1=r_1e^{i\theta_1}$ and ...
2
votes
3answers
47 views

$\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$

Show that $\sum_{j=0}^{n-1}z_j^k=\begin{cases} 0, & \text{if $1\leq k \leq n-1$ } \\ n, & \text{if $k=n$ } \end{cases}$, where $z_0,...,z_{n-1}$ are the $n$-th roots of unity. For $k=n$ it ...
0
votes
3answers
36 views

Problem with Proving a relation between Complex Numbers

Problem: If $$(1+i)(1+2i)(1+3i)...(1+ni)=x+iy$$Prove that $$2\times 5\times 10...(1+n^2)=x^2+y^2$$ $$$$ I'm really sorry but I have absolutely no idea as to how to start with this problem. It ...
2
votes
3answers
84 views

Prove $5^{31}13^{25}$ can be represent as $ a^2+b^2;a,b \in \mathbb{Z} $

Prove $5^{31}13^{25}$ can be represent as $ a^2+b^2;a,b \in \mathbb{Z} $ I read about complex numbers. Authors represent formula: $$({a_1}^2 + {b_1}^2)({a_2}^2 + {b_2}^2)\, = \,{({a_1}{a_2} - ...
2
votes
5answers
141 views

Solve an equation with complex numbers

The question is to solve the following equation for complex numbers $$z-i = iz +5$$ I have tried to add i to both sides which gives $$z = iz +5 + i$$ I have also tried with combinining all the terms ...
1
vote
2answers
28 views

Factoring a polynomial with complex coefficients

Given $$3z^2+6z+3i=0$$ Find the complex roots and write in the form $a+bi$. I want to see how to factor it when there is an $i$ being multiplied by the constant.
4
votes
1answer
69 views

Prove that this number is less than $1$,

a) Prove that, if $z$ and $w$ are complex numbers and $|w| = 1$, then $$\frac{|z-w|}{|1- \bar z w|} = 1$$ b) Prove that, if $|z|<1$, $|w|<1$, then $$\frac{|z-w|}{|1- \bar z w|} < 1$$ I ...
0
votes
1answer
33 views

Multiply two complex numbers

multiplication of two complex numbers - it's the same as multiplication of vectors. From physics i know that's result of multiplication of two vectors - it's a number. But when we multiply complex ...
1
vote
3answers
89 views

how did Cardano obtain three solutions for cubic?

So, if I am not mistaken Complex numbers were discovered after Cardano's method. But from Cardano's Method on Wikipedia, it says to get the three solutions, we should use the root of unity. In that ...
-2
votes
2answers
46 views

Basic Complex Number Questions [closed]

I have just started learning about complex numbers, so I would appreciate if any of you can show me the solutions to the following $2$ questions. Solve for $z$ and write your answer in rectangular ...
3
votes
4answers
77 views

Question about Euler's formula

I have a question about Euler's formula $$e^{ix} = \cos(x)+i\sin(x)$$ I want to show $$\sin(ax)\sin(bx) = \frac{1}{2}(\cos((a-b)x)-\cos((a+b)x))$$ and $$ \cos(ax)\cos(bx) = ...
0
votes
2answers
24 views

Find complex number $z$ if $arg(z^4i^{25})=arg(u)$ where $u=\frac{\sqrt{3}}{3}-\frac{1}{3}i$ and $|z|=6$

$\arg(u)=\frac{11\pi}{6}$ $t=(z^4i^{25})=2xy(y^2+xy-2x^2)+i(x^4-4x^2y^2+y^4)$ If $\arg(z^4i^{25})=\arg(u)$ does that mean $t=u$?
1
vote
0answers
32 views

How to deal with functions include complex number

If I have a complex number, say $e^{ix}=\cos x+i \sin x$. According to the definition of complex number, we know that the imaginary part of $e^{ix}$ is $\sin x$ and real part is $\cos x$. But if I ...
7
votes
6answers
111 views

A proper definition of $i$, the imaginary unit [duplicate]

Back when I was in high school, which was a long time ago, I recall my math teacher telling me that the definition of $i$, the imaginary unit, is $\sqrt{-1}$. Knowing little, at the time, I accepted ...