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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0
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0answers
24 views

Eigendecomposition of a complex symmetric matrix and the same with a shift on the diagonal

Suppose that the matrix $A\in \mathbb{C}^{n\times n}$ is complex symmetric matrix, thus is fulfills $A=A^T$ but also $A\neq A^H$, thus the matrix is not hermitian, but complex symmetric. Let's ...
2
votes
2answers
87 views

Why is the MacLaurin series proof for eulers formula $ e^{i\theta} = \cos(\theta) + i\sin(\theta) $ valid?

The proof for this $$ e^{i\theta} = \cos(\theta) + i\sin(\theta) $$ using the MacLaurin series is all right for a high school level, but I dont understand why the series that has been derived for the ...
2
votes
3answers
47 views

How to simplify $\Re\left[\sqrt2 \tan^{-1} {x\over \sqrt i}\right]$?

While solving $$\int \frac{x^2+1}{x^4+1}\,dx,$$ I tried to use partial fractions in the denominator by writing $x^4+1=(x^2+i)(x^2-i)$ And then I got $\Re\left[\sqrt2 \tan^{-1}{x\over \sqrt i}\right]$. ...
1
vote
1answer
35 views

Primitive 18-th root of unity problem involving congruences.

I have some doubts about this following problem, if you can please try to answer the congruence step: Let $ \omega$ be a primitive 18-th root of unity. Find $ n \in \mathbb Z$ such that: $ \omega^n =...
0
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1answer
21 views

Second order system - find -3dB frequencies and magnitude response analytically

Let's take some simple second-order system like $H(s) = \frac{j\omega T}{(1+ j \omega T)^2} $. I know that the magnitude response is simply the absolute of the function and the -3dB frequencies can be ...
0
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0answers
41 views

Prove or disprove: $\lambda\in\mathbb C$ is an eigenvalue of A $\implies\overline\lambda$ is an eigenvalue of A

I'm pretty sure it's true because every time I calculated eigenvalues for a complex matrix it eventually boiled down to solving a quadratic formula that gave me a conjugated pair, but not sure how to ...
3
votes
3answers
102 views

How can I calculate $(1+i)^{5404}$?

I saw a pattern while evaluating some other powers of similar complex number so I tried to calculate the above question by expanding it, please tell me if it is correct...? $(1+i)^{2} = 2i$ $(1+i)^{...
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1answer
25 views

Using the picture write down all values of $\sqrt[12]{z}$ and then find the main value of that number $z$.

Problem: Let $1+i\sqrt 3$ be one root of $\sqrt[12]{z}$. Display that number in complex plane and then, in that plane, display other roots of number $\sqrt[12]{z}$. Using the picture write down all ...
2
votes
1answer
122 views

Where does this equation come from? [duplicate]

Since I study 3 years i ask myself very often where does this equation come from? $$e^{i\theta} = \cos(\theta)+i \sin(\theta)$$ Is it found by series expansion?
-1
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2answers
53 views

Real Numbers Raised to Imaginary Powers? [closed]

What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples ...
1
vote
1answer
28 views

locus of complex number 2

Que: If $arg(\frac{z-z_1}{z-z_2})=\pi$ then what is the locus of $z?$ Doubt In my textbook it is written that it represents the straight line joining $A(Z_1)$ and $B(Z_2)$ but excluding the ...
0
votes
2answers
52 views

If $ f $ has pole at $0$ then show that $e^f$ can't have pole at $0$.

i am trying to show that if $ f $ has a pole at $0$ then $ e^f $ can't have removable singularity at $0$ ? I tried to show that but i have a problem . I assume that $e^f$ has removable singularity ...
0
votes
1answer
65 views

Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?

Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\...
0
votes
0answers
25 views

Circle and line construction of a compex number $z\in\mathbb C$

Let $C\subseteq\mathbb C$ be the field of constructible complex numbers; that is, it includes only the elements $z\in\mathbb C$ which can be constructed with circles and lines. The field $E\subseteq \...
1
vote
2answers
50 views

Is this simple looking complex expression valid always?

$$ z^{a+ib} = z^a*z^{ib} \hspace{2mm} \forall z\in \mathbb{C} $$ In high school I was always taught to see the + in complex numbers as analogous to that is reals. However can it be proven to be ...
-1
votes
1answer
19 views

Deriving hyperbolic functions

If $tan(\theta+i\phi)$=$sin(\alpha+i\beta)$. Prove that $sin2\theta*cot\alpha$=$sinh2\phi*coth\beta$. i approached by taking $tan(\theta-i\phi)$=$sin(\alpha-i\beta)$ and then i found $tan(\theta+i\...
0
votes
2answers
41 views

How to prove main argument formula for any $z\in\mathbb C^*$

I would prove that for any complex number $z \in \mathbb C^*$ such that $z = x + \mathbb i y$ with $(x,y)\in\mathbb R^2$ and $x+\vert z\vert \neq 0$: $$ \arg z = 2\arctan\left(\dfrac{y}{x+\vert z\vert}...
55
votes
12answers
8k views

Why is the complex plane shaped like it is?

It's always taken for granted that the real number line is perpendicular to multiples of $i$, but why is that? Why isn't $i$ just at some non-90 degree angle to the real number line? Could someone ...
2
votes
1answer
151 views

How does one show that $\cos {\left (\ln 2 \right )}\approx \frac{10}{13}$?

How does one approximate the value of something like this? Apparently Euler found the value of $\large \frac{2^i+2^{-i}}{2}\large $ [which equals $\cos {\left (\ln 2 \right )}$] to be close to $\...
0
votes
0answers
15 views

Prove that these pairs of complex numbers have real part 1/2 if they are symmetric in the complex plane.

Let matrix $A$ be defined as: $\Large A(n,k)=k^{-a_k + 1/2 + ib_k}$ if $k$ divides $n$, else $A(n,k)=0$ Let matrix $B$ be defined as: $\Large B(n,k)=\mu(n) n^{a_n+1/2 -ib_n}$ if $n$ divides $k$, ...
4
votes
1answer
63 views

Is it true that $ \sqrt{z^2-1} = i \sqrt{1-z^2} $?

I have seen a lot of times in books or on the internet that$ \sqrt{z^2-1} = i \sqrt{1-z^2} $ and I don't understand why that is correct . In general it is not true that $ \sqrt{-z}=i \sqrt {z}$ and ...
1
vote
2answers
41 views

Complex series should sum to zero but it's a puzzle

If we have a finite sum defined as $$\frac{1}{N}\sum\limits_{n=N/4}^{3N/4-1} e^{-4\pi ink/N}$$ (where $k$ is an integer and $N$ is divisible by $4$), then how can we show that this sum is equal to $...
1
vote
2answers
26 views

Magnitude of complex number $a=\frac {1-e^{-i\omega L}}{1-e^{-i\omega}}$

I tried using $\sqrt{a*a^*}$ but I still got some complex parts...shouldn't the magnitude contain no complex part? I did: $a*a^*=\frac {1-e^{-i\omega L}}{1-e^{-i\omega}}\frac {1-e^{i\omega L}}{1-e^{i\...
1
vote
2answers
48 views

What does it mean to perform calculus upon functions of complex values? [closed]

Complex numbers exist in a plane. This would lead me to believe that calculus views them as multivariate, but I am not real sure. How would one define a rate of change for a complex number valued ...
2
votes
4answers
92 views

A question about the definition of $\mathbb{C}$

In the usual definition of the field $\mathbb{C}$, as $\{(a,b):a,b\in\mathbb{R}\}$, the field $\mathbb{R}$ is not exactly a subset of $\mathbb{C}$, but only an isomorphic copy of the subfield $\{(a,0):...
2
votes
1answer
34 views

Logarithmic function in complex number [closed]

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.
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3answers
56 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 ...
0
votes
1answer
24 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?
4
votes
3answers
129 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
2answers
47 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 ...
3
votes
1answer
33 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
4answers
62 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 ...
1
vote
1answer
34 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? ...
14
votes
5answers
389 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\...
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= \...
2
votes
4answers
180 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+...
0
votes
1answer
27 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))...
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0answers
67 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 ...
3
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0answers
37 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
52 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
vote
1answer
80 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
55 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...
0
votes
2answers
74 views

High powers of complex numbers [closed]

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 ...
3
votes
0answers
100 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 ...
1
vote
1answer
21 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
32 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 ...
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{\...
2
votes
2answers
48 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 $$E := \{z\in\mathbb{C}:\vert z+i \vert = 2\vert z\vert \}$$ ...