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

0
votes
2answers
24 views

Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$

Problem : Let $z \in C^*$ such that $|z^3+\frac{1}{z^3}|\leq 2$ Prove that $|z+\frac{1}{z}|\leq 2$ My approach : Since : $(a^3+b^3)=(a+b)^3-3ab(a+b)$ $\Rightarrow ...
1
vote
1answer
26 views

Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of …

Problem : Let a,b,c be distinct non zero complex numbers with $|a|=|b|=|c|$ If each of the equations $az^2+bz+c=0$ and $bz^2+cz+a=0$ has a root having modulus 1, then prove that : ...
2
votes
1answer
74 views

Find the min and max distance from origin of the curve $\vert z+\frac{1}{z}\vert=a$

$z$ is a complex number, by the way. I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem. One of ...
1
vote
2answers
63 views

Solve $z^6+7z^3-8=0$

I want to find the solutions $z^6+7z^3-8=0$ but I don't know where to start because of the high degree of the equation. This is an exercise that involves complex numbers, so I have to transform the ...
0
votes
1answer
13 views

Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum…

Problem : Let a be a positive real number and let $M_a=\{z \in C^* : |z+\frac{1}{z}|=a\}$ Find the minimum and maximum value of $|z|$ when $z \in M_a$ My approach : $|z+\frac{1}{z}|=a$ Squaring ...
2
votes
4answers
35 views

Simplifying sum of powers of conjugate pairs

The result of summing a conjugate pair of numbers each raised to the power $n$: $$ (a + bi)^n + (a - bi)^n $$ Produces a real number where $a + bi$ is a complex number. Given the result is real, is ...
1
vote
2answers
25 views

Geometrical Description of $ \arg\left(\frac{z+1+i}{z-1-i} \right) = \pm \frac{\pi}{2} $

The question is in an Argand Diagram, $P$ is a point represented by the complex number. Give a geometrical description of the locus of $P$ as $z$ satisfies the equation: $$ ...
2
votes
2answers
1k views

Finding the least value for points in a loci

The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying ...
0
votes
1answer
29 views

Prove $f(z_0)I(\gamma;z_0)=\frac {g'(z_0)}{2\pi i}\int_{\gamma} \frac {f(z)}{g(z)-g(z_0)}dz. $

Let $f(z)$ and $g(z)$ be analytic in a region A and let $g'(z) \neq 0$ for all $z \in A$. Let g(z) be one to one and let $\gamma$ be a closed curve in A. Show that $$ f(z_0)I(\gamma;z_0)=\frac ...
0
votes
3answers
16 views

isolating x with two variables and negative exponents

I have: $$ 4^y = x^{-2} $$ Can someone hint to me what I need to do to isolate $x$? I'm not sure what to do.
1
vote
1answer
23 views

Higher degree polynomial with complex roots

I'm working on the following problem: $$ r^4 - 3r^2 -4r = 0 $$ I factor out one $r$ and leaving me $ r(r^3 - 3r -4) = 0 $. One real root is $r=0$, and I'm unable to find the other ones. I tried ...
0
votes
0answers
15 views

calculating $\int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2}$ using Residual Theorem [duplicate]

Could anyone help me provide a way to calculate $$ \int_0^{\pi} \frac {d\theta} {(a+b\cos \theta)^2} $$ using the Residue theorem in complex analysis? Many thanks
2
votes
2answers
165 views

Complex projective line homeomorphic to $2$-sphere

Define an equivalence relation $\sim$ on $X={\bf C}^2\setminus \{(0,0)\}$ by $(x_1,y_1)\sim(x_2,y_2)$ if and only if there exists $t \in C\setminus\{0\}$ such that $(x_1,y_1)=(tx_2,ty_2)$ show that ...
0
votes
2answers
36 views

Express $\sin^3x$ in terms of cosines of multiples of $x$

I am studying complex numbers, and I have been trying to figure that out. Just not getting it. I keep getting $\frac{1}{-i8 (2\cos(3x) - 2\cos(x) - i4\sin(x))}$.
2
votes
1answer
41 views

Under what conditions on $f$, is $f(az)=g(a)f(z)$?

Formal Statement Given nonzero constant $a \in \mathbb{C}$, $|a|>0$ and $f:\mathbb{C} \to \mathbb{C}$, under what conditions on $f$ does the following hold? \begin{equation} f\left(a ...
2
votes
0answers
83 views
+50

Multiplication of unitary matrices to make symmetric off-diagonal elements zero

Context Starting with a unitary matrix $U$ of size $m \times m$, I have read of a way to obtain a diagonal matrix by sequentially multiplying $U$ from the right by unitary matrices $V$ of a certain ...
379
votes
21answers
65k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
9
votes
5answers
2k views

Unexpected result from Euler's formula

I am a bit confused with a result I get from Euler's formula: $e^{2\pi i} = 1$ $\sqrt[3] { e^{2\pi i} }= \sqrt[3]{ 1 }$ $(e^{2\pi i})^{\frac{1}{3}}= 1$ $e^{\frac{2}{3} \pi i} = 1$ This last ...
4
votes
1answer
37 views

Graphically solving for complex roots — how to visualize?

So recently we've been doing the complex roots of quadratics, cubics and polynomials in general in school. But my question is, is there a way to see where these roots are, just like you can see where ...
1
vote
0answers
32 views

The singular points and residues of $\sin(\frac 1 z)$

I met a question asking all the singular points and corresponding residues of $$ \sin \frac 1 z $$ My understanding is that $$\sin \frac 1 z=\frac 1 z-\frac 1{3!z^3}+\frac 1 {5!z^5}+... $$ Thus ...
0
votes
3answers
62 views

Multiplication of real and complex radicals

If I have, for example, the product $\sqrt{7+\sqrt{22}}\sqrt[3]{38+i\sqrt{6}} $ Can I perform the multiplication or this cannot be done and only remains to leave the product in this form?
0
votes
1answer
34 views

An inequality with complex numbers.

Given $n$ complex numbers $z_1,\ldots,z_n$, is it true that $$ |z_j|\sum_{k=1}^n|z_k|\leq\sum_{k=1}^n|z_k|^2 $$ for $j\in\{1,\ldots,n\}$ ? Thank u for any help!
15
votes
6answers
825 views

Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$

Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
0
votes
0answers
17 views

Loops around 0 of polynomial restricted to the unit circle [duplicate]

Given a polynomial with coefficients in C, consider the image of the polynomial restricted to the unit circle (That is plugging in only things with absolute value one). How many loops around 0 can ...
0
votes
1answer
38 views

Complex Matrix Representation

Lets say if $X\in C ^{m\times n}$, it does have real and imaginary parts. If I want to represent a matrix in real and imaginary form then why I write it this way where is $i$? \begin{bmatrix} X_r ...
68
votes
9answers
4k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
0
votes
1answer
57 views

complex numbers and locus

When the problem says that the complex number $z$ moves on the straight line $y=2x$,what "clue" do I get from that? And generally when it says that a complex number belongs/moves to a conic section ...
1
vote
0answers
36 views

Proving analytic function $f = 0$ under certain assumtions

I was given the following exercise: Let $f(z)$ be analytic in an open and connected set $U$ containing the point $z=0$ and assume $|f(1/n)| < \frac{1}{2^n}$ for $n \in \mathbb{N}_{> 0}$. Prove ...
0
votes
2answers
29 views

How to find the absolute value of this complex number: $\frac{-4-6i}{17+i}$

I know that, in general, $|a+bi|=\sqrt{a^2+b^2}$, however, I don't know how to make $\frac{-4-6i}{17+i}$ into the form of $a+bi$.
0
votes
1answer
38 views

Solving an equation involving complex numbers.

I tried solving the problem on my own. I would like to know if I have made any mistakes. If I have indeed made a mistake, I would appreciate it if someone corrects it and explains what it is. Also, I ...
0
votes
0answers
18 views

Solving systems of linear equations with complex numbers by hand

How can I solve a 3x3 system of linear equations with complex numbers by hand without making a mistake? I know that I can solve them either with Gaussian Elimination or Cramer's rule, but I find it ...
6
votes
3answers
354 views

Convergence properties of $z^{z^{z^{…}}}$ and is it “chaotic”

$\DeclareMathOperator{\Arg}{Arg}$ Let $z \in \mathbb{C}.$ Let $b = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = {a_0}^{a_n}$ for $n \geq 1$, ...
2
votes
1answer
39 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle ...
1
vote
4answers
120 views

Real part of $\frac{1-e^{(n+1)i\theta}}{1-e^{i\theta}}$

How can I compute the real part of \begin{equation*} \frac{1-e^{(n+1)i\theta}}{1-e^{i\theta}}, \quad \text{where}\ \theta \in \mathbb{R}? \end{equation*} Maybe it's a silly question, but I'm feeling ...
3
votes
1answer
65 views

About prefactor in book's Gamma function identity

In "Mathematical Methods for Physicists" (Arfken & Weber, 7th ed.), exercise 13.1.16 says the following, Prove that $$|\Gamma (\alpha+i\beta)|=|\Gamma(\alpha ...
2
votes
1answer
308 views

Images of lines $y = k = \mbox{constant}$ under the mapping $w = \cos (z)$

I want to solve this question: find the images of lines $y = k = \mbox{constant}$ under the mapping $w =\cos(z).$ I know that $w=\cos(z)=\cos(x)\cosh(y)-i\sin(x)\sinh(y)$ so $u=\cos(x)\cosh(y)$ and ...
1
vote
1answer
54 views

Does $z^0$ Have Multiple Solutions?

While playing around with complex numbers, I stumbled upon a result that implies $z^0$ has infinitely many values (where $z$ is any complex number). This struck me as odd since I've never come ...
7
votes
1answer
215 views

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

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...
-3
votes
1answer
34 views

How to prove the Complex problem? [closed]

For any complex $z_0$,show that $$\lim_{z \to z_0} (z^2+1)=z_0^2+1$$
0
votes
1answer
37 views

How to prove $\lim_{z\to 0} \frac{z^2}{\overline{z}}=0$ using the definition of limit? [closed]

How to prove $$\lim_{z\to 0} \frac{z^2}{\overline{z}}=0$$? We can use definition of limit.
2
votes
1answer
46 views

Simplifications of nth roots of complex numers.

Is it easy to find that $( a + ib)^n$ is equal to a certain complex number, say $p+iq$, by just using Newton's binomial theorem. But how to find in general that $\sqrt[n]{p+iq} = a+ib$, where $a, b, ...
0
votes
0answers
21 views

Complex variable, multiplication of numbers

Question: Let a and b be complex numbers with $a \neq 0.$ Describe the set of points $az + b $ as $z$ varies over the first quadrant, $\{z = x+iy: x>0 \,and \,y>0\}$ Solution: Let $a = ...
1
vote
4answers
33 views

Complex numbers, finding solution for z

How can you solve this? $z^2+2(1+i)z=2+2(\sqrt{3}-1)i$ I have tried to compare left and right side with real and imaginary part i then get $ x^2+2x-y^2-2y=2$ $xy+x+y=(\sqrt{3}-1)$ But this ...
1
vote
1answer
56 views

Why don't we have $(\mathrm{cis}(2\pi))^{1/5} = (\mathrm{cis}(4\pi))^{1/5}$, while we do have $\mathrm{cis}(2\pi) = \mathrm{cis}(4\pi)$?

If $\operatorname{cis}(2\pi) = \operatorname{cis}(4\pi)$, then don't we have $$\big(\operatorname{cis}(2\pi)\big)^{1/5} = \big(\operatorname{cis}(4\pi)\big)^{1/5}?$$ This isn't yielding the same ...
2
votes
1answer
107 views

Find maximum value of$ |z_1 -z_2 |^2 + |z_2 -z_3 |^2 + |z_3 -z_1 |^2 $ if $|z_1 | = 2, |z_2 | = 3, |z_3 | = 4 $

i know $ |z_1 -z_2 |^2 = |z_1 |^2 + |z_2 |^2 - 2|z_1 | |z_2 | \cos \alpha $ where $ \alpha $ is angle between $z_1$ and $z_2 $. Similarly i get $ |z_1 -z_2 |^2 + |z_2 -z_3 |^2 + |z_3 -z_1 |^2 = ...
2
votes
3answers
98 views

How to reconcile the identity $\left( e^{i \theta} \right)^{1/2} = e^{i \theta/2}$ with the fact that $a^2 = b$ is solved by $a = \pm\sqrt{b}$

So I worked along the lines of the following: $$ \left( \cos \left( \theta \right) + i \sin \left( \theta \right) \right)^{\alpha} = \left( e^{i \theta} \right)^{\alpha} = e^{i (\theta \alpha)} = ...
0
votes
1answer
23 views

Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
-2
votes
3answers
54 views

Converting into rectangular form

I have 2 related questions: First: Let $z_1 = 2+2i$ and $z_2 = 2-2i$. Find $z_1z_2 $ in rectangular form. I have no idea... I'm also clueless about this question: Change the following to ...
2
votes
2answers
38 views

How to go from $\frac{1}{1+2j}$ to $\frac{1}{5} - \frac{2}{5}j$, where $j^2=-1$?

I am reading a book (DSP First), or mainly skipping through the pages trying to solve various exercises. At some point I came across this How exactly did we go from the second to the last step?
0
votes
0answers
15 views

Graphing complex numbers on Casio fx-9750 GII

I have a Casio fx-9750GII, I'm starting to get to grips with just how useful it can be. For those familiar with it you will be aware it can graph functions. However I noticed when looking through the ...