Tagged Questions

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

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-2
votes
2answers
35 views

Prove that $|\cos(iy)|>\frac{1}{2}e^{|y|}$

We know that for all complex numbers, $\cos^2z + \sin^2z = 1$. This doesn't apply that $|\cos z|<1$ and $|\sin z|<1$. Show that, for all $y \in \mathbb{R}$ $$|\cos(iy)|>\frac{1}{2}e^{|y|}$$ ...
0
votes
0answers
10 views

Images of some regions of the complex plane by given function?

I'm trying the draw the image of $A=\lbrace z\in \mathbb{C}:-1<Im((1+i)z)<1\rbrace$ by $f(z)=1/z$ and the one of $B=\lbrace z\in\mathbb{C}:|z|<1\rbrace$ by $f(z)=(z-1)^{-1}$. I've managed to ...
0
votes
0answers
42 views

Is there a rigorous proof for the existence of complex number?

Is there a rigorous proof for the existence of complex number? When I am learning analysis, there is a proof for the existence for irrational number. So I am wondering if there is a proof for the ...
1
vote
0answers
30 views

If $|z +\frac{1}{z}|=a$ find extreme values ​​of $|z|$

Let $a> 0$. Knowing that $z$ is complex number with $|z +\frac{1}{z}|=a$ find extreme values ​​of $|z|$. My partial solution: $$|z +\frac{1}{z}|=a <=> (z +\frac{1}{z})(\overline{z} ...
1
vote
1answer
10 views

Why left eigenvector complex conjugate transpose of right eigenvector?

My teacher today stated the following: For a matrix $A\in \Bbb R^{n \times n}$, any left eigenvalue $e^*$ is simply the transpose of the conjugate of a right eigenvector $e$ of $A$, so $e^* = ...
0
votes
3answers
59 views

$z^5=-2$ Complex number solutions

Find all complex number solutions to the equation $$z^5=-2$$ I'm a little lost in using De Moivre's theorem and Euler's formula.
5
votes
1answer
39 views

How to prove $\sum_{k=1}^{N} \frac{\sin n\theta}{2^N}=\frac{2^{N+1}\sin \theta + \sin N\theta -2\sin(N+1)\theta}{2^N(5-4\cos \theta)}$

Prove This using De Moivre Theorem $$\sum_{n=1}^{N}\frac{\sin n\theta}{2^n}=\frac{2^{N+1}\sin\theta+\sin N\theta-2\sin(N+1)\theta}{2^N(5-4\cos\theta)}$$ Please help me find my mistake, because ...
2
votes
3answers
60 views

Finding the roots of $\sec^2(x)=0$

I need to find the roots of $\sec^2(x)=0$ in my works. I know there are no real roots of this equation; are there complex roots?
4
votes
3answers
360 views

How to show that this complex equation has 10 non real roots and how to express them

I did the first part successfully: $$w^{12}=1= \cos 2\pi + i \sin 2\pi$$ $$w= \cos \frac{\pi k}{6} + i \sin \frac{\pi k}{6}$$ Where $k=0,1,2,3,4,5,6,7,8,9,10,11$ I struggled with this ...
-3
votes
0answers
17 views

plot the image of the unit circle under the complex mapping $f(z)=iz^3+z-i$ [on hold]

please I need help I need to do this on Matlab: a)plot the image of the unit circle under the complex mapping $f(z)=iz^3+z-i$ b)plot the image of the line segment from 1 to $1+i$ under the complex ...
5
votes
0answers
26 views

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. [duplicate]

If $f : D(0,1) \rightarrow \mathbb{C}$ is a function, $f^2$ is holomorphic, and $f^3$ is holomorphic, then prove that $f$ is holomorphic. MY ATTEMPT SO FAR: If $f^3$ is holomorphic, then we can ...
0
votes
1answer
22 views

complex number locus

The locus of the complex number Z is a rectangle in the Argand diagram with corners $(-a,0), (a,0), (a,a)$, and $(-a,a)$, where $a>0$. What is the locus of $Z^2$? It could be a relatively easy ...
1
vote
1answer
33 views

How to solve $(-19w + 93\overline w)^4=-1$

How to solve $(-19w + 93\overline w)^4=-1$ , if $w\in \mathbb C$ I really have no direction where to solve this question or at least a hint, can someone help?
2
votes
1answer
43 views

How do I solve $x^4+44=0$ according to de Moivre?

How do I solve $x^4+44=0$ according to de Moivre? I tried to use the formula, but I got roots that are not beautiful numbers. What should the complex roots for this equation be?
3
votes
0answers
88 views

$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
0
votes
0answers
12 views

Showing $z_1^{3k}+z_2^{3k}+z_3^{3k}=3(2^{3k}e^{\frac{5k\pi}{6}i})$

My Attempt First part: $$z=e^{(\frac{5\pi}{18}+\frac{2\pi n}{3})i}$$ Where $n=0,1,2$ Second part: $$z_1^{3k}= 2^{3k}e^{\frac{5k\pi}{6}i}$$ And so on, summing : ...
3
votes
1answer
47 views

solving for z in $|e^z| = 2$

How would I solve for z in the following case: $|e^z| = 2$, now I know that $|e^z| = e^a$ if we let $z = a+bi$ so then equating moduli we get $a = \ln{2}$ But what about $b$? $2 = 2e^{(0+2\pi n)i}$ ...
1
vote
1answer
58 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
1
vote
1answer
31 views

How to solve $e^{3z} = 1+i$

I am trying to solve $$e^{3z} = 1+i$$ Putting the RHS into modulus argument form, $$1+i = \sqrt{2} e^{(\pi/4 +2\pi n)i}$$ Now what I want to do is equate moduli and arguments, letting $z = a+bi$ ...
6
votes
1answer
93 views

Please help me find a complex number book suitable for me

Its been two weeks since I've joined this site, and I have received wonderful answers to my complex number questions at the shortest time. I am specially very weak in Complex numbers, and I see ...
1
vote
1answer
22 views

How to represent the solution of $z^{2}+2z+5=0$ in in Euler form?

I get the solution of $z^{2}+2z+5=0$ $z=-1+2i \;\; \overline {z}=-1-2i$
1
vote
1answer
37 views

Find an infinite sequence of numbers that can't be the derivatives of power series

First, I noticed that the nth derivative of $f$ at $0$ is $n!\cdot a_n$, but this does not really help me to construct a sequence of numbers that can not be generated by the sequence ...
1
vote
1answer
25 views

raising a complex number to a high power.

we should decide whether the following claims are right or not, and explain our decision. let $w_1,w_2,w_3$ be three different roots for the equation $z^3=1$ a) $w_1^{1991} + w_2^{1991} + ...
0
votes
0answers
20 views

Is there a simple way to solve this system of equations?

Is there a simple way to solve easily the following system of equations in the unknowns: $ x_2 , y_1 , y_2 , z_1 , z_2 \in \mathbb{C} $ depending on fixed values $​​a, b$ and $c$ in $ \mathbb{C} $ ...
1
vote
2answers
42 views

Modulus of z^-3?

What is the result of $|z^{-3}|$ and how can one show it? I know $z = e^{i\omega T}=cos(\omega T) + i\sin(\omega T)$, but I cant go further... I would be glad if someone can explain further.
0
votes
3answers
43 views

Given $z_1, z_2$ prove that $4z^2_1+9z^2_2 = 0$

I need to show that given $z_1 = 9 + 9i$ and $z_2=6-6i$, $$4z_1^2+9z_2^2=0.$$ $$z_1 = 12.7(cos 45 + i sin 45)$$ $$z_2 = 8.5(cos 315 + i sin 315)$$ I changed the terms to polar form, applied De ...
0
votes
1answer
17 views

Solve ${z_1/\overline{z_2}} = z^3$

$$z_1= 4\sqrt{2}-i4\sqrt{2}$$ $$z_2= \cos{135^\circ} +i\sin{135^\circ}$$ Find all the complex numbers $z$ that fulfill the following equation: $${z_1\over \overline{z_2}} = z^3$$ be aware that ...
0
votes
1answer
35 views

How to sketch the following set?

Sketch on the argand's diagram the following set: $A=\{w \in \mathbb{C} \backslash \{0\}:w^3-w^{-3} \in \mathbb{R}\}$. How to approach this question?
-5
votes
0answers
61 views

How would you answer these complex variable questions? [on hold]

$1.$ Let $u(x, y) = x \sin(x) \cosh(y) − y \cos(x) \sinh(y)$. Show that $u$ is harmonic and find a harmonic conjugate $v(x, y)$. Express $u(x, y) + iv(x, y)$ in the form $f(z)$. $2.$ Find all ...
0
votes
1answer
18 views

Sum of modulus of complex numbers

I'm trying to establish if |sin(z)|+|cos(z)| is greater than or equal to 1. I have tried to write out the expression in exponential form, but I don't really arrive at anything useful. I would really ...
4
votes
4answers
38 views

Conceptual question about the imaginary number $i$

One of the first things we see in our first complex analysis class is the standard way of introducing us to the imaginary unit $i$ which is to think of a solution to the equation $$x^2=-1$$ Obviously, ...
0
votes
1answer
46 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
0
votes
2answers
48 views

Show that complex numbers are vertices of equilateral triangle

1)Show if $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$ then $z_1,z_2,z_3$ are vertices of equilateral triangle inscribed in a circle of radius. I thought I can take use from roots of unity here, since ...
1
vote
1answer
161 views

Best way to solve $ z^{2} + i \cdot z = 0 $

What is the best way to solve $ z^{2} + i \cdot z = 0 $ ? I have tried to solve it via completing the square and using the quadratic formula, and got different answers.
1
vote
3answers
38 views

Show complex solutions exist

Let A be a complex number and B a real number. Show that the equation $\,\lvert z^2\rvert+ \mathrm{Re}\, (Az) + B = 0\,$ has a solution iff $\,\lvert A^2\rvert \geq 4B$. If this is so, show that the ...
0
votes
3answers
22 views

Are conditions equaivalent that they are roots of unity?

If I have conditions that $|z_1|=|z_2|=|z_3|=|z_4|=1$ and $z_1+z_2+z_3+z_4=0$ Is it suffice to state they are roots of unity ?
1
vote
3answers
30 views

Prove Complex Relationship

Prove that if z + 1/z is real then either the magnitude of z = 1 or z is real. Question i'm struggling with, I found that 2ab must be equal to 0 but I don't see how that help. Thanks!
1
vote
0answers
34 views

Proof about field extension : A geometric way

Let $M \subset \mathbb C $ be a sub-field which is not contained in $\mathbb R$ and which is closed under complex-conjugation. Let $L(M)$ be the set of all lines which crosses two points of $M$ and ...
2
votes
3answers
18 views

Show Trigonometric Identities from Complex indentity

So the exercise says to show $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and $\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)$ By using the following identity: $e^{i(a+b)}=e^{ia}e^{ib}$ How do ...
-1
votes
3answers
43 views

Why is the following equivalent transformation of the imaginary number legitimate? [duplicate]

Why is this substitution acceptable? $\sqrt{i}=\frac{1+i}{\sqrt{2}}$
3
votes
1answer
50 views

Riemann Zeta Function at $s = 1 + 2 \pi i n / \ln 2$

I am aware that the function defined by $$ \zeta(s) = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots $$ for $Re(s)>1$ can be extended to a function defined for $Re(s)>0$ by writing $$ ...
0
votes
1answer
37 views

Verify that $1+i = \sqrt 2 \cos(45^\circ) + i\sin(45^\circ)$ [on hold]

I need to verify that $$1+i = \sqrt{2}(\cos45^\circ + i\sin45^\circ).$$ Then I need to compute $(1+i)^{100}$. I've just learned this form for expressing complex numbers, so your help will be ...
2
votes
2answers
25 views

Find the Power Series

How would one write $f(z) = \frac{1}{1-wz}$ as a power series? ( Where $z,w$ are in $C$.) Would it just be $\sum_{n=0}^{\infty} (zw)^n$?
7
votes
1answer
50 views

Inequality relating coefficients and roots of a complex polynomial

While going through some olympiad handouts I stumbled upon a problem related to an upper bound for the Mahler measure, which stated that Given a polynomial $f(x) = x^n + a_{n-1}x^{n-1} + \dots + a_0 ...
0
votes
2answers
40 views

complex numbers

I have a number of questions about complex numbers and I need your help: z1, z2, z3, z4, z5 are complex numbers that fulfil |z1|=|z2|=|z3|=|z4|=|z5|=1 prove that |z1+z1+z3+z4+z5| = $|{1\over z1} ...
0
votes
2answers
29 views

complex numbers two problems

1) If $z=\cos\alpha+i\sin\alpha$ for $\alpha \in[0, 2\pi]$ then find $\alpha$ for $z^2+z$ I transform to this moment $\displaystyle ...
0
votes
2answers
30 views

Why is $t=\frac{1}{2}$ a root for $\tan 4\theta= \frac{4t-4t^3}{1-6t^2+t^4}=\frac{-24}{7}$, where $t=\tan \theta$

Show that $(2+i)^4=-7+24i$ $$\cos 4\theta = \cos^4 \theta - 6\cos^2 \theta \sin^2 \theta + \sin^4 \theta$$ $$\sin 4\theta= 4\sin \theta \cos^3 \theta- 4 \sin^3 \theta \cos \theta$$ ...
0
votes
1answer
15 views

z and w are two complex numbers prove the relationship

If $z$ and $w$ are complex numbers such that $|z+w| = |z-w|$ Prove that $\arg z - \arg w = \pm \ \pi/2$ Can someone please help me?
0
votes
1answer
9 views

finding equation of circle in complex plane

So i was asked to find the equation of the circle going through 1, i, and 0 Now i know that the equation of circle in complex form is: | z - (Zo) | = r where r is radius. So based on these values, ...
2
votes
1answer
55 views

How to find the roots of $(\frac{z-1}{z})^5=1$

Write down the fifth roots of unity in the form $\cos \theta + i \sin \theta$ where $ 0 \leq \theta \leq 2\pi$ Hence, or otherwise, find the fifth roots of i in a similar form By writing ...