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

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-4
votes
1answer
22 views

Find the priniciple argument of $-5\sqrt{2}+5i\sqrt{2}$ [on hold]

Find the priniciple argument of $-5\sqrt{2}+5i\sqrt{2}$ must be the exact value and in the form $z=...\pi$
0
votes
0answers
9 views

Is there a particular order when evaluating the complex exponential?

This question might seem silly. But I just cannot get this: $e^{i 6\pi} = 1+0i$ ($e^{i 6\pi})^{1/2} = \sqrt{1+0i} = \sqrt{1} = 1$ ($e^{i 6\pi})^{1/2} = e^{i3\pi} = -1+0i = -1$
-1
votes
1answer
41 views

If $\sin{x}+\sin{y}+\sin{z}= \cos{x}+\cos{y}+\cos{z}=0$, find the value of $\cos{2x}+\cos{2y}+\cos{2z}$. [on hold]

Is there any way to solve this question using complex numbers? I am trying the general way too but I am unable to solve the question.
1
vote
5answers
47 views

Complex Numbers Roots of Unity

By multiplying two roots, one is the conjugate of the other, we get one. Does someone know why and proof that. Many thanks
16
votes
5answers
1k views

Proving the following number is real

Let $z_i$ be complex numbers such that $|z_i| = 1$ . Prove that : $$ z\, :=\, \frac{z_1+z_2+z_3 +z_1z_2+z_2z_3+z_1z_3}{1+z_1z_2z_3} \in \mathbb{R} $$ This problem was featured on my son's final ...
0
votes
1answer
11 views

How to find the complex potential for the following flow under certain conditions?

We've used $z=i(Z+4/Z)$ as a conformal mapping to map the exterior of a circle $|Z|=2$ to the exterior of the line segment $(-4i,4i)$. We now want to write the complex potential of the uniform flow ...
0
votes
1answer
32 views

Hilbert's inequality for $\left|\sum_{n,m}a_n \bar a_m\right|$.

We know that, an Hilbert's inequality states $$\left|\sum_{n\neq m}\frac{a_n \bar{a}_m}{n-m}\right|\leq\pi \sum_n |a_n|^2$$ Give $a_n, b_n$ two sequences of complex numbers. Then write an inequality ...
7
votes
4answers
636 views

What is the motivation for complex conjugation?

I have been dealing with complex numbers for few years now. But when I've tried to think about the motivation behind complex conjugation, I was not sure. Let me write what I am working with. For a ...
1
vote
1answer
34 views

Find a Linear Fractional Transformations (LFT) $w(z)$

I have absulotly no idea how to approach this question, Can anyone please provide with a hint or any kinda information so I can solve this question. Thank you very much for you help
-2
votes
2answers
42 views

Cauchy-Riemann equations Complex Numbers [closed]

I have used the theorem if f'(z) = 0 then f(z) is a constant. I have proved it by using Cauchy Riemann's theorem. b
3
votes
2answers
34 views

Complex Differentiation

Can anyone give a hint to how to approach this question?
4
votes
3answers
168 views

Find$\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 2x + 4}\,dx$ and $\int_{-\infty}^{\infty} \frac{\sin(x)}{x^2 + 2x + 4}\,dx$

Find $$\int_{-\infty}^{\infty} \dfrac{\cos(x)}{x^2 + 2x + 4}\,dx$$ and $$\int_{-\infty}^{\infty} \dfrac{\sin(x)}{x^2 + 2x + 4}\,dx$$ I find it really difficult. Much appreciate it if anyone can ...
-3
votes
0answers
14 views

use De Moivre's theorem to show that: cos(3theta) = 4cos(3theta)-3cos(theta) and sin(3theta) = 3sin(theta) - 4sin(3theta)

let z = cos(theta) + isin(theta) a) find z^3 using binomial expansion b) use De Moivre's theorem to show that: cos(3theta) = 4cos(3theta)-3cos(theta) and sin(3theta) = 3sin(theta) - ...
0
votes
0answers
49 views

how can I show that $\cot\pi z$ and $\csc \pi z$ have simple poles for every integer $n$? so then I can calculate residues at those poles?

how can I show that $\cot\pi$z and $\csc\pi$z have simple poles for every integer $n$? so then I can calculate residues at those poles?
1
vote
1answer
29 views

Separate imaginary and real parts from complex expression

I learned about complex numbers after I was trying to create a fractal object. The main problem is that I have an equation with complex numbers and I have to separate their parts (real & ...
0
votes
1answer
35 views

Find the maximum & minimum value of complex number.

Let $z_1, z_2, z_3, \ldots, z_{13}$ be real numbers, & let $A$ be the average of complex numbers $[e^{iz_1}, e^{iz_2}, \ldots ,e^{iz_{13}}]$, where $i=\sqrt{-1}$. As the value of z's vary over ...
3
votes
0answers
78 views

Matrix representation of $\mathbb{C}$ as $^*$Algebra.

We know that there are many matrix representations of the field $\mathbb{C}$. For $2 \times 2$ real entries matrices, e.g., all the subrings of $M(2,\mathbb{R})$ generated by $I$ and a matrix $J$ such ...
2
votes
2answers
45 views

Are complex differentiable function in a point analytic?

I know that if a function $f$ is complex differentiable in a neighborhood of $z_0$, then we say it's holomorphic in $z_0$ and it's also analytic in a neighborhood of $z_0$. But suppose that I know ...
3
votes
4answers
45 views

Simplifying quartic complex function in terms of $\cos nx$

$$z= \cos(x)+i\sin(x)\\ 3z^4 -z^3+2z^2-z+3$$ How would you simplify this in terms of $\cos(nx)$?
1
vote
1answer
24 views

Taking the complex conjugate of some complicated composite function

I'm aware of the rule where to take the complex conjugate of anything, you simply replace any $i$'s with $-i$, and to conjugate any composed functions (i.e. $f*(g(z))=f(g*(z)))$ What is the ...
0
votes
0answers
20 views

Where does $\cos z$ conjugate is holomorphic

I solved this question using Cauchy-Riemann equations and got a contradiction,meaning not analytic everywhere, but I am not sure I am right. Got from 1 equation $x=\pi k$ from the other $x=\pi/2+\pi ...
1
vote
0answers
26 views

How find set $f(\Bbb R^3)$ for $f(x, y ,z)= e^{i(x+y+z)} + e^{i(x-y-z)}+ e^{i(-x+ y-z)}+ e^{i(-x-y+ z)}$?

Let $f:\Bbb R^3 \to \Bbb C$ such that $f(x, y ,z)= e^{i(x+y+z)} + e^{i(x-y-z)}+ e^{i(-x+ y-z)}+ e^{i(-x-y+ z)}$. How can one find the set $f(\Bbb R^3)$?
0
votes
2answers
31 views

Radius of convergence of a complex power series question

I know the general idea for ROC (radius of convergence) is to use the ratio test, or lim sup, however how would i go about solving this?
1
vote
0answers
59 views

Problem with multivaluedness of $(-1)^{\frac 14}$

Assume that $p\equiv3\mod4$ is an odd prime and $k$ an odd number. Then $$m=(-1)^{\frac{p^k-p^{k-1}+2}{4}}$$ seems to be always the value $1$ (?). This would be interesting how one can prove this - I ...
0
votes
1answer
23 views

Finding if a group is a vector space

$\mathbb{C}^2$ is a group over field $\mathbb{C}$, with the following actions: addition is similar to the regular addition. multiplication is defined by: for every $(z,w) \in \mathbb{C}^2$ and every ...
1
vote
0answers
24 views

compact convergence for a series in complex space

I need some help with this. I have to show that the follwing series converges compat. $$\sum_{n=1}^\infty f_n :D:= \{z \in \mathbb{C} | Re(z) > 0 \} \to \mathbb{C}, f_n (z):=\frac{1}{z+n^2} $$ I ...
6
votes
7answers
210 views

Solving complex numbers equation $z^3 = \overline{z} $

We have the following equation: $$z^3 = \overline{z} $$ I set z to be $z = a + ib$ and since I know that $ \overline{z} = a - ib$. I was trying to solve it by opening the left side of the equation. ...
0
votes
1answer
15 views

Let $z_1$ and $z_2$ be the $nth$ roots of unity which subtend a right angle at origin, then prove that n must be of the form $4k$

Problem : Let $z_1$ and $z_2$ be the $nth$ roots of unity which subtend a right angle at origin, then prove that n must be of the form $4k$ Solution : Here $arg \frac{z_1}{z_2}=\frac{\pi}{2}$ ...
2
votes
4answers
65 views

How do you solve $\cos \pi z =0$?

How do you solve $\cos \pi z =0$? I am unsure what to do with the $\pi$. I know how to solve $\cos z = 0$, but $\pi$ is throwing me off. Can someone help start me off with this question please?
3
votes
8answers
214 views

What is the value of $i^0$?

I have to solve the following question - $$\sum_{n=0}^{1000} i^n$$ where $i = \sqrt{-1}$ To be able to solve the problem, I need to know the value of $i^0$. What is the value of $i^0$? Is it 0 or ...
1
vote
0answers
45 views

complex nos in ellipse.

I was practising some ques on ellipses when I came a criss this question: If normal at four points $(x_1,y_1)$..... on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ are concurrent then find the ...
2
votes
4answers
134 views

solution for complex number equation with power

i have the following equation $$z^3 = 2 + 2i$$ and I need to find the solutions of this equation in trigonometric form. how do I find multiple solutions of such an equation?
0
votes
4answers
39 views

Finding a Trigonometric Form of Complex Number

I need to find the trigonometric form of the complex number: $1-i\sqrt{3}$ I found that $r = 2$ which means the trigonometric form is $2 ( \cos \alpha - i\sin \alpha)$ and I need to find the ...
1
vote
1answer
21 views

getting the absolute value of complex numbers

How can I get the absolute value of the following complex number such as $|(1-i\sqrt{2})^3|$ ? what is right way to solve it?
1
vote
2answers
93 views

A report about complex numbers

I was told to make a report for mathematics, and I could choose my own subject. I chose complex numbers, because I really think they are interesting. However, my teacher says that there isn't a lot of ...
1
vote
0answers
32 views

Two-term asymptotic approximation for roots of a polynomial (dominant balance)

I'm trying to find the roots to the following equation: $t^5 - \epsilon t^3 + \epsilon^3 = 0$ as $\epsilon \rightarrow 0$. From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + ...
1
vote
2answers
65 views

Is the following complex number “finite”?

This is my first question on this forum, so, forgive me in advanced if I make some type of syntax error... I am working on applying a theorem which involves computing a definite integral, and showing ...
4
votes
0answers
33 views

The image in $\mathbb{C}$ of $\mathbb{R}^2$ under a map of counterpropagating plane waves is…?

Define $$f_n(\mathbf{r})=\frac{1}{n}\sum_{k=1}^n\exp\left(2\pi i\binom{\cos\left(2\pi k/n\right)}{\sin\left(2\pi k/n\right)}\cdot\mathbf{r} \right)$$ as the sum of $n$ counterpropagating plane waves. ...
6
votes
3answers
111 views

Does $1^{\frac{-i\ln 2}{2\pi}}$ equal 2?

Just out of curiosity, I would like to know if this derivation is correct or not. Let's assume complex numbers and write $1 = e^{2\pi i n}$, for any $n\in\mathbb{Z}$. Then, by exponentiation we ...
3
votes
3answers
48 views

Roots of unity are distincts

For every $n\in\Bbb N$ and $$z_{k}:= \cos(2\pi k /n)+i\sin(2\pi k /n), \qquad k = 0,\ldots,n-1$$ we have $z_k^n=1$. How to show, in a simple way, that $z_k\neq z_l$ for every $k\neq l$? By ...
3
votes
3answers
36 views

Finding real coefficients of equation given that $a+ib$ is a root

Below is the question present in a past examination paper. I'll be giving my attempts and how I thought it through. Do feel free to point out any mistakes I make throughout my working even if ...
4
votes
1answer
86 views

Subset $A$ of $\mathbb{C}$ such that $\prod_{a\in A}(1+a)=1$

Let $A$ be a finite subset of $\mathbb{C}$ with at least two elements such that $f:z\to z^2$ induces a bijection from $A$ to $A$. If $1\notin A$, how can I show that $\displaystyle\prod_{a\in ...
3
votes
1answer
71 views

Finding $\prod_{k=1}^{n-1}\cos\frac{2k\pi}n$

Finding $$\mu=\prod_{k=1}^{n-1}\cos\frac{2k\pi}n$$ I thought $$z^n=1=e^{i2\pi}\implies z=\cos\frac{2k\pi}n+i\sin\frac{2k\pi}n\quad k\in\{1,2,...,n-1\}$$ Now we have: ...
1
vote
3answers
66 views

Solutions to $z^3 - z^2- z =15 $

Find in the form $a+bi$, all the solutions to the equation $$z^3 - z^2- z =15 $$ I have no idea what to do - am I meant to factor out z to get $z(z^2-z-1)=15$ or should I plug in $a+bi$ to z? ...
1
vote
3answers
68 views

Distinct roots of $z^n-z$

How would we prove that for any positive integer $n$ the complex roots of $z^n-z$ are all distinct? In the case that $n=1,2,3$ I have factored it directly but how can we do it in general?
26
votes
8answers
820 views

Why is $1/i$ equal to $-i$?

When I entered the value $$\frac{1}{i}$$ in my calculator, I received the answer as $-i$ whereas I was expecting the answer as $i^{-1}$. Even google calculator shows the same answer (Click here to ...
0
votes
1answer
27 views

Complex Coefficients and Real roots

Find $m$ which is a real number so that this equation has a real root. $2z^2-(3+8i)z-(m+4i)=0$ I've tried $b^2-4ac=0 $ but I can only seem to get complex $m$ values, so either I'm missing a key ...
0
votes
1answer
26 views

Prove that the line integral on $\beta$ of $f'(z)/f(z) = (A-B)/2 \pi i$ using Rouche's Theorem

Suppose that $\alpha$ is a regular closed contour. $f$, our function, lacks zeros and poles on $\beta$ and if A=the number of zeros of f inside $\beta$ (a zero of order n is counted n times) and B= ...
0
votes
1answer
24 views

If $p>0$ demonstrate that the $1/2\pi i$ the line integral of $z^p f'(z)/f(z)$ is $\sum (z_k)^p$

This is basically a deviation of Rouche's Theorem from what I can tell. My first instinct was to do this via induction in which we know that $p=0$ we would have Rouche's theorem. But it gets ...
2
votes
1answer
27 views

Question about finding Laurent Series over closed region and classifying singularity

Represent $\sin(\pi x/(x+1))$ Laurent Series about the region $0<|x+1|<2$: Its true that $$\sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ So the $$\sin(\pi x/(1+x))=\sum (-1)^{n-1} \frac{(\pi ...