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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4answers
41 views

What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane?

What is the area of the triangle having $z_1$, $z_2$ and $z_3$ as vertices in Argand plane? Is it $$\frac{-1}{4i}[z_1(z_2^* - z_3^*)-z_1^*(z_2-z_3)+{z_2(z_3^*)-z_3(z_2^*)}]$$ where $w^*$ denotes the ...
0
votes
2answers
34 views

Argument of complex numbers

If $z=re^{i\theta}$ and $w=\rho e^{i \phi} $ are two complex numbers, then $ arg(zw)=arg (z)+arg (w)$ But if $z=-1$ and $w=-1$, we get $ 0= 2\pi $ which is not correct. So why it gives us this ...
7
votes
1answer
340 views

Why does the Mandelbrot shape show up in other fractals?

In the pictures below, the Collatz map fractal includes parts resembling the Mandelbrot set. Why? Do other fractals do so? The Mandelbrot set From Wikimedia Commons Part of the Collatz map fractal ...
2
votes
1answer
29 views

Logarithmic function in complex number

Show that: $$\cos[i\log(2+\sqrt3)]=2$$ I attempted by taking$(2+\sqrt3)$ into trigonometrical form but i am stuck Please help me out.
0
votes
3answers
37 views

A geometric approach to this problem?

Question: A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z$, where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane ...
4
votes
3answers
120 views

What is the geometric interpretation of $|z-1|^2+|z+1|^2=4$ for all $z$ such that $|z|=1$?

Show that $|z-1|^2+|z+1|^2=4$ for all z such that $|z|=1$. [Note that $|z|$ refers to the magnitude of z where $z=a+bi$]. I was able to 'prove' the question; however, I cannot think of a geometric ...
0
votes
1answer
20 views

What is a geometric interpretation of multiplication/division in the complex plane? [duplicate]

How can one visualize the multiplication/division of a complex number, z, by a real number, an imaginary number, or another complex number?
3
votes
1answer
29 views

What is the kernel of $\phi$?

Let $\phi: \mathbb{C}^* \to \mathbb{R}^*$ with $z \mapsto |z|$ be a homomorphism. What is the kernel of this homomorphism? We know the identity in $\mathbb{R}^*$ is $1$. So we need to find the ...
1
vote
1answer
33 views

Show the limit exists.

For $|z|\neq 1$,show that the following limit exists: $$f(z)=\lim_{n\to\infty}\frac{(z^n -1)}{(z^n+1)}$$ Is it possible to define f(z) when $|z|\neq 1$ in such a way as to make $f$ continuous? ...
2
votes
2answers
66 views

Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$

my question is too simple. We know all that if we define the exponential function on $\mathbb{C}$ then we define the real part and imaginary part of $\exp{it}$ as $\cos{t}$ and $\sin{t}$. So if we ...
1
vote
1answer
35 views

Write $\,-4i\,$ in polar form

Write $\,-4i\,$ in polar form ${re}^{i\theta}$, with $r$, $\theta\in \mathbb R$, and $\,r\geq0,\;0\leq\theta<2\pi$. I let $\,z=-4i\,$ first, then get $\,r=\sqrt{0+{4^2}}=4$. However, $\,\tan\theta\...
2
votes
4answers
169 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
1
vote
3answers
175 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
0
votes
0answers
10 views

Modulus of complex number arrangement

Hithere, I have solved my equation to a point where I have: $$z_1 z_2 = 8r cis\frac{5\pi}{4}$$ $$\vert z_1 z_2 \vert = 2$$ Would I be correct in saying the modulus is 8r, thus $8r = 2$, so $r= \...
0
votes
1answer
24 views

Maximal value of real part of holomorphic function

Let $f:U \rightarrow C$ be a non-constant holomorphic function. $U$ is open, connected and $D(0,1+\epsilon) \subset U$. I'd like to show that there exists $z_0 \in \partial D(0,1)$ such that $Re(f(z))...
1
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0answers
63 views

Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics. First there were natural numbers. People started adding things and finding solutions to finding the ...
384
votes
20answers
66k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
2
votes
0answers
24 views

Proof check: commutation of Galois automorphisms and complex conjugation in CM-fields

Let $K/\mathbb{Q}$ be a Galois CM-field with $Gal(K/\mathbb{Q})=:G$ and $J_\mathbb{C}$ be the complex conjugation. Since $K$ is a CM-field one can show, that $$J:=\phi^{-1}\circ J_\mathbb{C}\circ \phi=...
3
votes
2answers
50 views

Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right) $$ where I taken ...
1
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3answers
2k views

For complex $z$, find the roots $z^2 - 3z + (3 - i) = 0$

Find the roots of: $z^2 - 3z + (3 - i) = 0$ $(x + iy)^2 - 3(x + iy) + (3 - i) = 0$ $(x^2 - y^2 - 3x + 3) + i(2xy -3y - 1) = 0$ So, both the real and imaginary parts should = 0. This is where ...
1
vote
1answer
68 views

Is it true that $ \sqrt{-z} = i \sqrt z $?

Is it correct to write $ \sqrt{-z} = i \sqrt z $ , for every complex $z$? I think it's not true but I have seen it in some books . The reason I think it's not correct is for example if $z=i$ then $\...
1
vote
2answers
54 views

Calculating the gcd of complex numbers

I need help in calculating the gcd of complex numbers For Example: $\gcd(3+i,1-i)$. The problem is,I don't even know what's the algorithm for complex numbers...
3
votes
0answers
69 views

Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
0
votes
2answers
74 views

High powers of complex numbers [on hold]

I have these two questions that I am trying to solve. I know that I am suppose to use De Movire's Theorem but I am getting stuck. Can you guys please help out? Thanks. Compute the following ...
5
votes
4answers
280 views

Easy partial fraction decomposition with complex numbers

There is an easy method to perform a partial fraction decomposition - described here, under the "Repeated Real Roots" title, for the coefficient A2. The problem is ...
1
vote
1answer
18 views

Inequality of absolute value of a complex number

If $ z $ is a complex number, does it follow that $ |z| \ge z $ like with real numbers? The way I justify it is by saying that if $ z \in \mathbb{C}, $ then $ z = a +bi $ for some $ a,b \in \mathbb{R}....
0
votes
3answers
31 views

Triangle inequality for complex numbers

I just start to learn about complex numbers and I want to prove the triangle inequality, which says that if $ z $ and $ w $ are complex numbers, then $ \displaystyle |z + w| \le |z| + |w|. $ My ...
0
votes
1answer
35 views

Need help finding conjugate

$$\overline{z-2+4i} = 2z+3+8i$$ I got this question on my online assignment. I got to a point where I couldn't get rid of the conjugate of z and I don't know how to expand or what to do with it. I ...
2
votes
2answers
47 views

How to sketch the region on the complex plane? [duplicate]

I am going through a basic course on complex analysis. I have a problem in understanding the following. E $\subset\mathbb{C}$ is defined as $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ ...
1
vote
1answer
35 views

graphing a circle in the complex plane? [on hold]

The ellipse seemed rather simple: Defining the equation of an ellipse in the complex plane But Wolfram won't graph it with equal axes. http://www.wolframalpha.com/input/?i=abs{%28x%2Biy%29}%2Babs{%...
1
vote
2answers
50 views

Automorphism of unit disk without zero

Let $S$ be the unit disk without $0$. Find all $f \in Auto(S)$ I got the following idea. By Riemann 0 is a removable singularity. Since for $g\in Auto(D)$ where $D$ is the unit disk. $g(z)= e^{i{\...
0
votes
3answers
38 views

How are the following factors 'linear'.

What does it mean for factors to be linear? Q: Find the four linear factors of: $$z^4+z^3+z^2+z+1$$ I got the following: $$(z-e^{i \pm {2\pi \over 5}} )(z-e^{i \pm {4\pi\over 5}} )$$ I though ...
-1
votes
2answers
39 views

Computing $(3+a_1)(3+a_2) \ldots(3+a_n)$ [on hold]

If $n+1$ is an odd positive integer and $1,a_1,a_2,\ldots, a_n$ are the $(n+1)^{th}$ roots of unity, then $(3+a_1)(3+a_2) \ldots(3+a_n)=$ ?
2
votes
1answer
71 views

How to simplify $\sqrt{-8}$

How would I go about simplifying square root of $-8$? I know I can rewrite that as $\sqrt{(-1)(8)}$, and then I would get $i\sqrt{8}$, but how do I simplify that $8$ further? Thanks for your help.
-2
votes
0answers
19 views

What is the value of Z2? [on hold]

Z1,Z2,Z3 are vertices of an equilateral triangle circumscribing the circle |Z|=1, if Z1=1+i√3 and Z1,Z2,Z3 are in anticlockwise sense, then Z2=?
11
votes
4answers
1k views

Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial, (degree 4) either can have 0 real, 1 real, 2 real, or 4 real roots, and the rest is complex roots. Why can it not have 3 real roots and 1 complex?
0
votes
6answers
87 views

Homework quesiton: Find $p$ and $q$ so that $(p+qi)^2=3-4i.$

I got as far as taking the square root of both sides, and I'm ashamed to say that I'm already stuck. Any pointers? In regards to the comment, I got as far as $q^4+3q^2-4=0$ by equating the parts that ...
3
votes
1answer
109 views

Finding all $z\in \mathbb{C}$ such that the series $\sum\limits_{n=1}^{\infty} \frac{1}{1+z^n}$ converges

I am trying to find out all $z\in \mathbb{C}$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{1}{1+z^n}$ converges. I notice that for $\left|z\right|\leq 1$, we have $\left|1+z^n\right|...
0
votes
1answer
67 views

What is the benefit of representing a complex number as e^i(theta) versus e^(a+bi), what is the process of finding a solution to this example?

What is the benefit of representing a complex number as $ e^{i\theta} $ versus $ e^{a+bi} $? Am I correct in saying that these give the same information but offer convenience in different situations? ...
0
votes
3answers
56 views

How to derive values for $i$ raised to negative integers? [closed]

This link states that the values of $i$ raised to the power of negative integers. How can we derive these values from the positive powers?
2
votes
2answers
47 views

Taking Mod on both sides, mathematically correct?

When given a equation containing complex numbers such as $$ \frac{a+ib}{c+id} = x + iy$$ and required to prove $$ \frac{a^2 +b^2}{c^2+d^2} = x^2 + y^2$$ Is taking the mod of both sides a legal ...
1
vote
3answers
75 views

Proof Involving Imaginary Number: Where's the wrong one? [duplicate]

Here are the propositions: $$i=\sqrt{-1}$$ $$i^2=-1$$ $$(i)(i)=-1$$ $$\sqrt{-1}\sqrt{-1}=-1$$ $$\sqrt{(-1)(-1)}=-1$$ $$\sqrt{1}=-1$$ There's an error in the propositions above. I think it's in the ...
0
votes
3answers
36 views

modulus and argument of $(-4\sqrt{3}-4i)^3$?

Any fast method to obtain the modulus and argument of $(-4\sqrt{3}-4i)^3$? If i use the exponential form to solve it, is it good?
1
vote
1answer
38 views

minimum of $F(z) = 4\left|z-1-2i\right|^2+13\left|z-4-5i\right|^2+12\left|z-2-7i\right|^2+23\left|z-6i\right|^2$

Find Minimum value of $F(z) = 4\left|z-1-2i\right|^2+13\left|z-4-5i\right|^2+12\left|z-2-7i\right|^2+23\left|z-6i\right|^2$ $\bf{My\; Try::}$ Put $z=x+iy\;,$ We get $$f(x,y) = 4(x-1)^2+4(y-2)^2+...
-1
votes
1answer
17 views

Can you help me to understand this Magnitude/Phase to Real/Imaginary conversion?

I've a module/function that takes an array of magnitudes/phases and get to me the real/img results. These are the input values. Magnitude values: ...
12
votes
1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
0
votes
2answers
42 views

A sum of powers of primitive roots of unity

For the primitive roots of unity $\omega_n = e^{i2\pi/n}$ I want to prove that $$\sum_{k=0}^{n-1} \omega_n^{lk} = 0$$ if $n$ doesn't divide $l$. I have already proven the well-known result $$\sum_{k=...
1
vote
2answers
40 views

Help with De Moivre's Theorem: Complex Numbers

I have a homework problem which goes: Given $z^n=(z+i)^n$, using de Moivre's Theorem, show that $z=\frac{i}{e^\frac{i2k\pi}{n}-1}$ What steps should I take in tackling this question? It's a 2 mark ...
-1
votes
1answer
65 views

Image drawing complex analysis [closed]

$w=u+iv,z=x+iy$ are complex numbers and we have $w=z^2-2z$. Determine the image in the $w$-plane of the unit circle $x^2+y^2=1$. I have tried to answer this here Question and Answer. I have problems ...
1
vote
2answers
41 views

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$, if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$