Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

learn more… | top users | synonyms

1
vote
3answers
90 views

Describe the locus of complex numbers $a=\lambda_1z_1+\lambda_2z_2$ where $\lambda\in\mathbb C$

Suppose $z_1, z_2$ be any two fixed points in the complex plane and let $\lambda_1, \lambda_2$ be two real numbers such that $\lambda_1+\lambda_2=1.$ If $a$ be the complex number such that ...
1
vote
1answer
147 views

how to solve complex integration problem

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. The only idea that I am getting is that we can use the ...
1
vote
1answer
67 views

Estimating the modulus of the roots of $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $$\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$$ lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
1
vote
2answers
62 views

Find 2 imaginary numbers that have a cosine of 4, using $\cos z =\frac{e^{iz}+e^{-iz}}{2}$

Use the definition $$ \cos z =\frac{e^{iz}+e^{-iz}}{2} $$ to find $2$ imaginary numbers having a cosine of $4$. I tried two approaches, both of which ended in failure: $$ 8=e^{iz}+e^{-iz}\\ ...
1
vote
3answers
674 views

Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.

Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real. Your answer should be expressed as a set of the form $S = \{z \in\mathbb C : \text{conditions satisfied by }z\}$. ...
1
vote
1answer
77 views

Complex conjugate of polar form of $z \in \mathbb C$

Express in regular form the conjugates of $z \in c$ that satisfy $z^2 + 4i = 0$ Let $$ z=re^{i\theta} $$ Thus $z^2 = -4i = 4e^{i\frac{3\pi}2} $ $r^2=4, r=2$ $\theta_1 = \frac{3\pi}4, \theta_2 = ...
1
vote
1answer
24 views

How to solve $z^3=\frac12-\frac12i$

After simplifying the equation: $(2i)^9z^3=(1+i)^{17}$ I got at the end that: $z^3=\frac12-\frac12i$ but I didn't know how to take it from here, and how to find $z_1$, $z_2$, $z_3$.
1
vote
2answers
34 views

How to solve higher grade polynomials of complex numbers $q^{10}-2q^5+2=0$

If I wanted to find the roots for $q^{10}-2q^5+2=0$, how would I go about doing that? I tried treating it like a quadratic equation, but couldn't get there. I also tried putting $q=(a+ib)$ but that ...
1
vote
2answers
41 views

Complex Cubic Equation z^3+3z+2i=0

How we can solve the equation $z^3+3z+2i=0$ ? And is there exist a general method to solve similar equation?
1
vote
1answer
96 views

Is the series: $\,\sum_{n=1}^{\infty}\frac{\mathrm{e}^{-in}}{n}\,$ divergent?

According to mathematica, the complex series $\displaystyle\sum_{n=1}^{\infty}\frac{e^{-in}}{n}$ does not converge. I know that the factor $\dfrac{1}{n}$ in the above series is diverging, but I don't ...
1
vote
1answer
165 views

Find square roots of $8 - 15i$ [duplicate]

Find the square roots of: $8-15i.$ Could I get some working out to solve it? Also what are different methods of doing it?
1
vote
2answers
69 views

Complex power of a real number

What is the meaning of $(-1)^{i}$, where $i^{2}=-1$ and what is its value?
1
vote
1answer
63 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...
1
vote
4answers
143 views

Show that if $z,w\in\mathbf{C}$, $|z|<1$ and $|w|<1$, then $\left|\frac{z-w}{1-\overline{z}w}\right|$<1? [duplicate]

Every way I try to approach this turn it into proving the inequality $|z-w|<|1-\overline{z}w|$. Not sure at all how to approach it at the moment.
1
vote
2answers
594 views

What does the Fraktur-R or Re stand for in math?

I have an assignment where we are suposed to find the angle between two complex vectors and we have been given the formula to try to work out the problem and were told by the prof. that he got it ...
1
vote
3answers
126 views

A tricky complex numbers if and only if proof

For complex numbers $z$ and $w$ prove that $$|z|^2w -|w|^2z = z-w\quad \iff\quad z=w\quad\text{or}\quad z\bar{w}=1.$$ How would you go about solving this problem?
1
vote
3answers
88 views

Complex Equations

The Equation: $$ z^{4} -2 z^{3} + 12z^{2} -14z + 35 = 0 $$ has a root with a real part 1, solve the equation. When it says a real part of 1, does this mean that we could use (z-1) and use ...
1
vote
3answers
110 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
1
vote
4answers
138 views

If $ w_1 $and $ w_2 $ are the roots of $w^2-2sw+t=0$, then $|w_1|=|w_2|\iff 0<s^2/t\le1$

Let $s$ and $t$ be two complex numbers not equal $0$, $ w_1 $and $ w_2 $ are solutions to this equation $$w^2-2sw+t=0$$ How can prove this equivalence ? $$|w_1|=|w_2|\iff 0<\dfrac{s^2}{t}\le1$$. ...
1
vote
2answers
85 views

Generating all the Pythagorean triples by factorizing using complex numbers

Can anyone help me generate all the triplets solution of the Diophantine equation: $a^2+b^2=c^2$ by factoring using Complex numbers? thanks.
1
vote
1answer
46 views

Complex variable algebra mishap

One question on a problem set was the following: Show that $x^2 - y^2 = 1$ can be rewritten as $z^2 + \bar{z}^2 = 2$. (With $z = x + iy$) So I started working from the first expression based on ...
1
vote
2answers
108 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...
1
vote
3answers
68 views

Determining Laurent Series expansion and residues

Determining Laurent Series expansion and residues of $f(z)=\frac{z}{(z+1)(z+2)}$ around $z = -2$. What is the validity of the expanded region? What is $res(f, -2)$??
1
vote
3answers
866 views

Express a complex number in modulus amplitude form

Express a complex number in modulus amplitude form $\displaystyle 1+\sin \alpha +i\cos \alpha $ My Attempt: $\displaystyle r\cos \theta= 1+\sin \alpha $ $\displaystyle r\sin \theta= \cos \alpha $ ...
1
vote
2answers
70 views

Sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1 \}$

My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. I substituted $z$ for $a+bi$, but did not get anywhere: ...
1
vote
3answers
39 views

Related to the construction of $\Bbb C$ (generalisation)

To construct $\Bbb C$, we consider $\Bbb R^2$ endowed with the operations: $$\begin{align} (a,b) + (c,d) &:= (a+c, b+d) \\ (a,b) \cdot (c,d) &:= (ac - bd, ad+bc)\end{align} $$ then write ...
1
vote
2answers
84 views

Trigonometric equation with complex numbers

Let $x$, $y$, and $z$ be real numbers such that $\cos x+\cos y+\cos z=\sin x+\sin y+\sin z=0$. Prove that $\cos 2x+\cos 2y+\cos 2z=\sin 2x+\sin 2y+\sin 2z=0$. Starting with the given equation, I got ...
1
vote
2answers
42 views

$z=(-1+i)^{11}+(-1-i)^{15}=?$

Can someone help me in this question : Let $z=(-1+i)^{11}+(-1-i)^{15}$ so $z=-96+160i$ $z=96-160 i$ $z=160-96i$ $z=-160+96i$ what is the right answer ? Thanks in advance.
1
vote
3answers
152 views

Laurent series and residue of $f(x)=\frac{1}{1+e^z}$

I am having trouble trying to expand this function using Laurent series, and finding the residue$$f(x)=\frac{1}{1+e^z}$$ If I replace $e^z$ with its series I get ...
1
vote
3answers
62 views

Proof using complex numbers

Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right| = 1$ where $\bar{z}$ is conjugate of $z$ and $\bar{z}w\ne 1$ if either $|z| = 1$ or $|w| = 1$. I used $|c_1/c_2| = |c_1|/|c_2|$ and multiply out with ...
1
vote
5answers
67 views

Explain why there are two complex numbers z such that $|z| = 1$ and that satisfy the equation $|z| = |z-1|.$

I must find both such complex solutions and express them in Euler form and usual form. So it's been a while since I've touched the imaginary/real plane. However, from what I remember, $z = a + bi$. ...
1
vote
3answers
94 views

Prove a function is not continous

How can I (formally correct) prove that $\;f: \mathbb{C} \rightarrow \mathbb{C}$ = $ \left\{ \begin{array}{cl} 0 & z = 0 \\ e^{-\frac{1}{z^2}} & z \neq 0 \end{array} \right.$ is not ...
1
vote
1answer
87 views

Finding argument of a complex number

How do you evaluate the following $$\text{Arg}\{\sin\frac{8\pi}{5} + i(1 + \cos\frac{8\pi}{5})\}$$
1
vote
2answers
58 views

Limit concept under Complex analysis

Prove that $$ \lim_{z\to i} \dfrac{3z^4-2z^3+8z^2-2z+5}{z-i} = 4+4i $$
1
vote
2answers
114 views

What is $\tilde{\Bbb{C}}$

$\tilde{\Bbb{C}}$ was defined in the following manner $\tilde{\Bbb{C}} = \Bbb{R} \cdot 1 + \Bbb{R} \cdot e$ with $1 \cdot 1 = 1, 1 \cdot e = e \cdot 1, e \cdot e = 1$ Could you elaborate more on ...
1
vote
2answers
208 views

Determine all complex numbers z in equation:

Let $n\in\mathbb{N}$. Determine all complex numbers $z\in\mathbb{C}$ such that $z^{n-1}$ = $\bar{z}$ How would I begin this? Would I begin by saying $z=a+ib$ and expand and stuff?
1
vote
2answers
85 views

Cauchy-Schwarz in complex case, using discriminant

There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz ...
1
vote
2answers
53 views

ODE with complex char roots gives strange solutions

$y''-4y'+5y=0$ has char roots - $\{e^{(2+i)x},e^{(2-i)x}\}.$ So its solutions is $e^{2x}\cos(x), e^{2x}\sin(x).$ But when i plug, e.g., first of them into original eq. i get: $-4 e^{2x} cos(x) + 8 ...
1
vote
2answers
176 views

Roots of Unity - Complex Numbers

The sets $A = \{z : z^{18} = 1\} $and $ B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. ...
1
vote
2answers
442 views

express $\mathrm{e}^{(2+i \pi/2)}$ in form $a + bi$

I'm just starting out into Complex numbers, polar and exponential form etc... I can happily convert numbers such as $\mathrm{e}^{i \pi/2}$ but I'm a little stumped with how to handle the extra + 2 ...
1
vote
1answer
401 views

Complex numbers - exponential numbers - (double angles?)

I am half stumped on this rather confusing problem: Let x, y, and z be real numbers such that $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$. Prove that $\cos 2x + \cos 2y + \cos 2z = \sin ...
1
vote
2answers
219 views

Proving entire function constant

$f$ is entire and $f(z)=f(z+1)=f(z+i)$ prove $f(z)=const$ I have no clue how to solve it
1
vote
4answers
367 views

factor $z^7-1$ into linear and quadratic factors and prove that $ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$

Factor $z^7-1$ into linear and quadratic factors and prove that $$ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$$ I have been able to prove it using the value of $\cos(\pi/7)$. Given here ...
1
vote
2answers
85 views

complex numbers - Proving

Prove algebraically that $|w+z|\le|w|+|z|$ for any complex numbers w and z. This is what I got so far: Since $|z|^2 = z \overline{z}$, we square both sides: $ |w+z|^2 = (w+z) \overline{(w+z)} \le ...
1
vote
2answers
96 views

How to solve a complex equation $w^4 = \sqrt{3} -i$

$z = \sqrt{3} -i$ How do I solve a complex equation $w^4 = \sqrt{3} -i$ I know that I first have to rewrite z to polar format which I have done as $z = 2(cos (-π/6) + sin (-π/6))$ but I do not know ...
1
vote
1answer
39 views

${{a+b\sqrt-7}\over{2}}$ as the root of a polynomial

So, I need help in showing that for any integers $a$ and $b$, ${{a+b\sqrt-7}\over {2}}$ is the root of a quadratic polynomial with integer coefficients if and only if $a$ is congruent to $b (mod 2)$. ...
1
vote
1answer
1k views

Solving the differential equation $y'' + 2y' + 2y = 0$ given constraints

How can I solve this initial value problem? $$ y'' + 2y' + 2y = 0,$$ given $y\,(\pi/4)=2$ and $y'(\pi/4)=0$. I've found $y(t)=e^{-t} \left(C_1\cos t + C_2\sin t \right)$ but I wasn't able to find ...
1
vote
2answers
49 views

Make the vector $[1,1]$ turn of an angle - $\pi/4$ , with complex numbers

We have $[1,1]$ and $\theta = -\pi/4$ here is my attempt: $(\cos(-\pi/4) + i \sin(-\pi/4)) * (x+iy)$ = $(\sqrt{2}/2 - i \sqrt{2}/2) (1+i)$ = $\sqrt{2}/2 - i^2\sqrt{2}/2 $ = $[\sqrt{2}/2 + ...
1
vote
3answers
70 views

Complex number: Roots

Solve all the roots of the following equation: $$(z-i)^2(z+i)^2=\frac{1}{4}.$$ Find the set of complex numbers $z$ such that $$\left|\frac{z-3}{z+3}\right|=2.$$ Would anyone mind telling me how ...
1
vote
2answers
453 views

What does $i^i $ equal and why? [duplicate]

I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - Is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator ...