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

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5
votes
7answers
202 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
64 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
213 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
130 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
92 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
64 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
107 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 ...
3
votes
1answer
84 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? ...
-4
votes
1answer
30 views

Roots of Unity; Complex Numbers [closed]

How do you find roots of unity? Are roots of unity different from roots of complex numbers?
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
811 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 ...
0
votes
1answer
40 views

A problem in understanding principal root in the complex plane.

We know that every complex number has exactly $n$ $n$-th roots in the complex plane, and we usually take (if the context where we are working doesn't tell us more) the one with real and imaginary part ...
0
votes
1answer
26 views

Application of Rouche's theorem on $x^4-6x+3$

I'm being asked to find the number of zeros given $|z|<1$ and $1<|z|<2$. So here are my inequalities but I'm not quite sure how to find the number of zeros though based on these inequalities. ...
3
votes
1answer
57 views

Prove that the integral of $x\cos(x)/(x-2)(x-1)$ from negative to positive infinity is $\pi(\sin1-2\sin2)$. Use an indented contour

To do this I used the Residue Thm but the main issue here is that I cannot get the sine term to appear. Perhaps I'm ignoring something here. We know that the singularity is $x=1,2$ so we should just ...
0
votes
1answer
33 views

Expand $(e^{2x}-1-2x)/x^5$ into Laurent Series on 0<|x|<$\infty$ and classify its singularity

I guess I'm having difficulty with this because its not in the form of a polynomial expression, which is what I've been taught. Nevertheless here's what I did: I know that the expansion for ...
0
votes
1answer
34 views

Complex number, power series

Develop $\sinh z$ in powers of $z-\pi i$ to show that $$\lim_{z\to \pi i}\frac{\sinh z}{z-\pi i}=-1$$ I know that $\sinh z=\sum_{n=1}^\infty \frac{z^{2n-1}}{(2n-1)!}$. Edit: Following the hint ...
1
vote
3answers
30 views

Complex number, entire function

Let $f(z)=\frac{(e^{cz}-1)}{z}$ if $z\neq0$ and $f(0)=c$ show that f is entire Theorem:A power series represents a analytical function inside their circle of convergence. I know I could prove ...
2
votes
3answers
68 views

Prove that the integral of $\sin^2(x)/(5+3\cos(x))$ from $0$ to $2\pi$ is $2\pi/9$

I'm not really unsure of how to approach this problem. I was thinking of reparametrizing the sin and the cos to its exponential form but I realize that it becomes even messier and leads sort of ...
0
votes
4answers
63 views

Complex numbers equivalence proof

I stumbled upon this exercise and can't seem to have any fruitful attemps, or rather I can't put together missing links. Let $z_1, z_2, z_3$ be different complex number such that $|z_1|=|z_2|=|z_3|$. ...
0
votes
1answer
26 views

Evaluate the integral of $e^{x}/(x+1)^4$ on $\rho$, which denotes the entire imaginary axis

I'm not entirely sure if my intuition is correct but the singularity for this equation is -1 but -1 does not exist on the imaginary axis, so does this integral equal 0? If not, what am I missing and ...
-1
votes
1answer
37 views

Prove that the integral of $sin(z)/(z^2+4z+5)$ from negative to positive infinity is $-\pi sin(2)/e$

I think I've made the problem a lot nastier than it supposed to look. Here's what I have so far. First notice that $(z^2+4z+5)$ is equivalent to $(z^2+4z+4)+1$ so our singularities are -2-i and ...
0
votes
1answer
35 views

Complex Series proof

Integrate the Maclaurin series for$\frac{1}{1+z}$ along a path, inside the circle of convergence, going from $z'=0$ to $z'=z$ and show that $$Log(z+1)=\sum_{i=1}^\infty (-1)^{n+1}\frac{z^n}{n}, ...
0
votes
2answers
70 views

Prove that $\int_0^{\infty} \frac{x^2}{x^4+5x^2+4}dx = \frac{\pi}{6}$

Prove that $\int_0^{\infty} \frac{x^2}{x^4+5x^2+4}dx = \frac{\pi}{6}$ Obviously you would use Residue Theorem to tackle this problem. The correct answer to this is $\frac{\pi}{6}$ however I'm ...
1
vote
1answer
32 views

Minimum value of $\lvert z_1-z_2\rvert $ given $\lvert z_1-i\rvert ^2=4$ and $\lvert z_2-6\rvert =\lvert z_2\rvert $

Minimum value of $\lvert z_1-z_2\rvert $ given $\lvert z_1-i\rvert ^2=4$ and $\lvert z_2-6\rvert =\lvert z_2\rvert $ The answer is supposed to be $1$, but I keep getting $0$ when I graph the problem. ...
5
votes
4answers
189 views

Imaginary fraction square root?

I have a fraction - $-\frac{1}{3}$ Which could either mean the value of fraction is $\frac{-1}{3}$ or $\frac{1}{-3}$ Note the minus sign Now, what is the sqaure ...
1
vote
2answers
64 views

How to calculate $\zeta(i)$?

As the title says, I'm interested to know how $\zeta(i)$ is calculated. I know the functional equation for the zeta function, but if I put it in that in there, I must know $\zeta(1-i)$. Is it a good ...
0
votes
0answers
17 views

I always have some doubts regarding the inequalities in cases where the function become Complex in the field for the real numbers

Consider this inequality $x + \log\left(x \right)> \log\left (x\right) - 2$ Does this inequality has $-1$ as its solution ? It will be very helpful for me.
0
votes
3answers
86 views

How to simplify if $a > 0$ and $\cos(a) < 0$ [closed]

$$\sqrt{\cos (a)} \sinh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia}\right)}\right)+\sqrt{\cos (a)} \cosh \left(\ln (2) a^{\frac{1}{2} \left(e^{-ia }+e^{ia }\right)}\right)=$$
1
vote
4answers
91 views

Simplify your answer completely: $5i(1 + i)^2$

Simplify your answer completely: $5i(1 + i)^2$ I know the answer is $-10$, but I don't know how to get it. Things I tried: foiling $(1+i)^2$ and then distributing $5i$. distributing $5i$ into ...
1
vote
4answers
41 views

A seemingly simple property of complex numbers that won't submit.

Let $a,b\in\mathbb{Z}[i]$ such that $|b-a|>|b|$ and $|a|>|b|$. I want to show that the absolute value of the real part of $\frac{a}{b}$ is greater than $\frac{1}{2}$. For example, let ...
3
votes
1answer
24 views

Determinant of skew-hermitian matrix

Given a skew-hermitian matrix $A \in \mathbb{C}^{N\times N}$, then $A = -A^H = -(A^*)^{T}.$ We can also say that $A^T = (-(A^*)^T)^T = -A^*.$ Thus, when computing the determinant we get $$ \det(A) = ...
1
vote
3answers
51 views

Cubic root of unity

Is there anyway to solve this without substituting with the values? Prove that: $$\frac{1+10w^2}{1-2w} + \frac{2+17w}{2+3w} = 6$$. (Where $w$ & $w^2$ are the cubic roots of unity)
0
votes
1answer
17 views

Euclidean circle in complex plane

I am reading Anderson's Hyperbolic Geometry and am having trouble with one of the Exercises in Chapter 1: Consider the unit circle $\mathbb{S}^1=\{z \in \mathbb{C} \text{ s.t. }|z|=1\}$. Let $A$ be a ...
-7
votes
1answer
33 views

help with complex numbers question [closed]

Add, subtract, multiply, divide,conjugate and plot complex numbers and modulus on an argand diagram. Evaluate: $(3+4i)+(4-2i)$ Evaluate: $(-3-4i)-(2-3i)$ Evaluate: $(2+2i)\times(3-3i)$ Plot the ...
1
vote
3answers
25 views

Nonlinear system with complex numbers

Solve the following system under the complex numbers (without eulerian form or polar form) $$z^3 + w^5 = 0 \\ z^2 \bar w^4 = 1$$ I have found that $(\pm 1, \mp 1)$ satisfy the equations as well as ...
0
votes
1answer
15 views

Algorithm for finding Complex Eigenvectors?

I'm wondering if there's a fairly easy algorithm by which one can, by hand, find eigenvectors corresponding to complex eigenvalues for small matrices. Of course, one can always row reduce, but it can ...