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
1answer
34 views

Is the dimension $-1$ the real $0$th dimension and does this all make sense?

I know there are at least two questions on this site that ask about the negative dimensions. But I want to ask something more than that. We have a number line. It contains all the real numbers we can ...
6
votes
1answer
186 views

$m+ni+k\lambda,\,\Re(\lambda),\Im(\lambda)\notin \mathbb{Q}$ is dense in $\mathbb{C}$!

As said in the comments below, it's needed to suppose $\{1,\Re(\lambda),\Im(\lambda)\}$ linearly independent over $\mathbb{Q}$, otherwise the result is false, according to Christian's example. ...
1
vote
2answers
40 views

Determine the largest open set to which $f(z)=\sum_{n=1}^{\infty}(-1)^n(2n+1)z^{n}$ can be analytically continued

Let $U=B_1(0)$ and $$f:U \rightarrow \mathbb{C},\qquad f(z)=\sum_{n=1}^{\infty}(-1)^n(2n+1)z^{n}.$$ Determine the largest open set to which $f$ can be analytically continued Remark: I was given ...
4
votes
0answers
28 views

Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
2
votes
1answer
53 views

What do $\int_{-1}^1\frac{dx}{2x+1-2i}$ and $\frac12\log(2x+1-2i)$ mean?

Suppose we want to evaluate $$I=\oint_C\frac{dz}{z+\frac12}$$ where $C$ is the unit square with diagonal corners at $-1-i$ and $1+i$. If we let $z:=re^{it}-\frac12$, then ...
22
votes
3answers
2k views

Can I compare real and complex numbers?

I'm calculating the eigenvalues of the matrix $\begin{pmatrix} 2 &0 &0& 1\\ 0 &1& 0& 1\\ 0 &0& 3& 1\\ -1 &0 &0 &1\end{pmatrix}$, which ...
-3
votes
1answer
38 views

Formula Derivation

What is the formula that can be derived using the values and formulas below to get the value of K43? Here's what I have, so far, but no luck in getting the correct formula: 0 = 925191 - 119355 - ...
1
vote
1answer
27 views

An inequality for complex number $|a+b|^p \sim |a|^p+|b|^p$.

I know that for any nonnegative numbers $a,b$ and $1\leq p<\infty$ then $a^p+b^p\leq (a+b)^p\leq 2^{p-1}(a^p+b^p)$. Now we need to find the similar inequalities for complex numbers. My question ...
2
votes
3answers
151 views

$(-1)^{\sqrt{2}} = ? $

This popped up when I was thinking about $$(-1)^{\frac {p}{q}}$$ where $ p $ and $q$ are integers such that $\gcd (p,q) = 1$ If $p$ is even : $(-1)^{\frac {p}{q}} = +1$ If $q$ is even : ...
2
votes
2answers
45 views

Prove that $\max_{|z| = 1} |P(z)| \ge 1$

I got stuck on this problem: Given a polynomial on complex plane $P(z) = z^n + a_{n-1}z^{n-1} + ... + a_1 z + a_0$ for $z \in \mathbb{C}$. Prove that $\max_{|z| = 1} |P(z)| \ge 1$ What I tried ...
0
votes
3answers
27 views

Complex Numbers and Euler/Polar Form

Say you have a complex number with $|z|=2$ and argument of $-\pi/3$. Why is it not valid to say $e^{-\pi/3i} = e^{5\pi/3i}$? Is it still valid to say $2cis(-\pi/3) = 2cis(5\pi/3)$?
0
votes
1answer
31 views

Powers with complex/negative bases

If x can be a positive real number (for example a fraction with a numerator and denominator), then why does the following relationship hold true only if and only if a and b are strictly positive real ...
0
votes
3answers
34 views

determinant of SU(3) matrix

I don't understand the determinant condition on SU(3) group, broadly. I know that the determinant of such matrices should be equal to 1. But what is the real intention of that 1? Is it the real ...
0
votes
3answers
76 views

Solutions of $z^5 = -1$

I have found the solutions to $z^5=-1$ but I have to use the following factorization to find the complex number produced when all solutions are multiplied. Each solution is denoted by $z_0-z_4$: ...
0
votes
1answer
69 views

Are all numbers expressible as a complex number? [closed]

Are there any numbers that are not elements of the complex field? Follow-up questions: Are p-edic fields subfields of the complex field? Can quaternions be viewed as a complex vector space in three ...
0
votes
2answers
58 views

Finding the 5th root of a complex number

I've got myself really confused on this question, and I'm not sure where to go from my attempt. Thanks for any help! QUESTION: Find the 5th roots of $1 + 2i$. ATTEMPT: I've began by finding the ...
0
votes
3answers
143 views

$\sqrt{x}=-1$. How can I solve it?

I am so curious about this equation: $\sqrt{x}=-1$ Does the $x$ where $x\in \mathbb{C}$ exist? How can I solve it?
2
votes
1answer
107 views

Showing $3<\pi<2\sqrt{3}$ using complex analysis

First of all we define $\pi$ to be $\pi=2\sup\{t\geq 0 :\; \text{for}\; 0\leq s \leq t, \Re(e^{is})\geq 0 \; \text{and}\; \Im(e^{is})\geq 0\}$ And we know that $e^{\frac{\pi}{2}i}=i$,$e^{\pi i}=-1$ ...
0
votes
1answer
10 views

Are G-numbers equivalent to Eisenstein integers?

In 100 Great Problems of Elementary Mathematics the author terms the set of numbers $$xO+yJ:x,y\in\mathbb{Z}$$ where $J=-\omega^2$, $O=-\omega$ and $\omega=e^{2\pi i/3}$ as G-numbers. These are used ...
1
vote
0answers
22 views

Relation between the Complex Number System and Vector Spaces

Given that any Complex Number $z$ can be represented as a Vector in $\mathbb{R^2}$ and since a Vector is nothing more than an element of a Vector Space (in its most general form), does that not imply ...
0
votes
1answer
48 views

Hyperbolic Trigonometry with Complex Numbers

I was trying to show the following complex hyperbolic trigeometric relation ...
1
vote
0answers
38 views

Q: A quadrilateral, equilateral triangles and two perpendicular lines

I'm trying to solve the following question using complex numbers: Let $ABCD$ be a convex quadrilateral with the equilateral triangles $ABE,BFC,CGD,DHA$ constructed externally on its sides. Prove ...
2
votes
2answers
48 views

How many elements are in the set $ \{ \left( \frac{2+i}{2-i} \right) ^n : n \in \mathbb N\}$

How many elements are in the set $$ \left\{ \left( \frac{2+i}{2-i} \right) ^n : n \in \mathbb N\right\}$$ My attempt: $\left( \frac{2+i}{2-i} \right) ^n = \left( \frac{3}{5} + \frac{4}{5}i ...
1
vote
1answer
23 views

complex number multiplication by a real number [closed]

I'd like to multiply a complex value by a real integer. I know that multiplication of complex numbers is similar in the polar form, but the way I know and have been taught is to multiply the two real ...
0
votes
3answers
90 views

Complex numbers proof…

For any two complex numbers Z and W. Prove that: $$|Z|^2W-|W|^2Z=Z-W$$ if and only if: $$Z=W \quad\text{or}\quad ZW^* = 1$$ Use the upper equation to derive any one of the lower equations.
0
votes
4answers
46 views

The number $2$ is a real root of $z^3=8$. Find the distance $|2-w|$ if $w$ is a complex root of $z^3=8$?

Question: The number $2$ is a real root of $z^3=8$. Find the distance $|2-w|$ if $w$ is a complex root of $z^3=8$ leaving your answer in surd form. What I have attempted: $$z^3=8$$ $$ ...
5
votes
3answers
102 views

Prove that $\oint_{|z|=r} {dz \over P(z)} = 0$

I got stuck on this problem, hope anyone can give me some hints to go on solving this: P is a polynomial with degree greater than 1 and all the roots of $P$ in complex plane are in the disk B: ...
1
vote
1answer
33 views

Find the smallest $n\in\mathbb N_{\geq1}$, such that $\left(\frac {z_1}{z_2}\right)^n\in\mathbb R$

I've been trying to solve this problem in multiple ways but can't seem to find a reasonable solution. We've got two complex numbers: $z_1 = a+bi$ and $z_2 = c+di$. Say $\lambda = \frac{z_1}{z_2}$. ...
0
votes
1answer
25 views

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$?

How to show that $\{x,y,,z\}$ are linearly independent $\Rightarrow$ $\{x+y,x+z,y+z\}$ is independent does not hold for arbitrary field $F$? I was thinking about polynomial space and complexe space ...
0
votes
0answers
17 views

Two different random processes for generating polynomials

Consider two processes for generating random complex polynomials: choosing the roots uniformly and independently throughout the unit disc, and choosing the coefficients uniformly and independently ...
0
votes
0answers
12 views

How to solve/calculate this equation with complex arguments

I understand that In($3$)=$x$ means $e^x=3$, where e=$2,71828$...But I don't know how to calculate/solve In($i$) and In($2+3i$). Can you tell me how to calculate/solve this? (though it's not unique)
0
votes
2answers
57 views

When is $\sin\colon\mathbb{C}\to\mathbb{C}$ purely real/imaginary?

Sketch the sets on which $\sin\colon\mathbb{C}\to\mathbb{C}$ is purely real/imaginary. My current result is that for purely real numbers $\sin$ is purely real and for purely imaginary numbers ...
1
vote
1answer
41 views

Examples of complex-variable functions that fail to have a limit at some point

My notes from class have the example $\frac{\overline{z}}{z}$ as z tends to zero. That the limit does not exist is shown by exhibiting that along the $x$-axis the limit is $1$ and along the $y$-axis ...
2
votes
3answers
52 views

Real and imaginary parts of $\cos(z)$

Not sure if I have done this correctly, seems too straight forward, any help is very appreciated. QUESTION: Find the real and imaginary parts of $f(z) = \cos(z)$. ATTEMPT: $\cos(z) = ...
0
votes
2answers
38 views

Prove the following equality

So I have the following equality involving complex numbers: $$\frac {\sqrt 3 -1}{1-i}(1+\sqrt 3 \,i)(\cos \alpha -i\,\sin\alpha)=2\sqrt {2-\sqrt 3}\left(\cos\left(\frac {7\pi}{12}-\alpha\right)+ ...
0
votes
1answer
15 views

Interior Uniqueness: Does there exist an analytic function on a neighborhood of $z=0$ that satisfies the following?

I am faced with the following problem: Does there exist a function that is analytic on a neighborhood of $z=0$ and satisfies the following condition for every positive $n$: (a) ...
-2
votes
0answers
38 views

Prove that $\sum_{k=1}^{n}z_k = \frac{n}{2}z_0 \ \ (n \ge 4)$ in the following case.

Let $z_0 ,z_1 ,\ldots,z_n $ be complex numbers such that $$|z_k -z_0 | = r \text{ , for } k = 1, 2, 3, \dots, n$$ and $$\begin{cases}\arg z_1 &=\frac{2\pi}{n}\\[4pt] \arg \left( {z_{k+1}}/{z_k} ...
0
votes
4answers
70 views

What region in $\mathbb{C}$ does $\left|{z-1}\right|+\left|{z+1}\right|$ = 2 describe?

I have played around with this a bit and keep getting something that doesn't seem right. Perhaps I'm overlooking something. Using the definition of distance in the complex plane I transform my ...
1
vote
0answers
21 views

Integral around a square in the complex plane

Let $f(z)$ be any continuous function defined in the complex plane with the property that $$\bigg|\int_{R_n}f(w)dw\bigg|\leq n^2\log(n),$$ for any $n>1$ and any square $R_n$ with side length $n$. ...
1
vote
1answer
34 views

Find Maclaurin series for integral of $e^{z^2}$

I need to find a Taylor Series expansion of $\displaystyle \int_{0}^{z}e^{\zeta^{2}}d\zeta$ around $z=0$, which shouldn't be hard enough. Except that I can only integrate term-by-term if the Taylor ...
0
votes
0answers
40 views

How to find the common denominator with multiple variables

Find $\frac{zf^{'}(z)}{f(z)}$, where $-1 \leq \alpha \leq 0 $ and $0< v < 1$ Given: $f(z)= \frac{1}{\pi}(-\log (1-vz)+ \alpha \log(1-vz^{-1}))$ and $f^{'}(z)= \frac{1}{\pi}\left(\frac{v}{1-vz} ...
3
votes
2answers
54 views

Solving $\lvert z \rvert z^2 = \sqrt{2}(1-i)\overline{z}$

I haven't been able to solve the complex equation $$\lvert z \rvert z^2 = \sqrt{2}(1-i)\overline{z},$$despite trying writing $z$ in different forms and using $z\overline{z}=\lvert z\rvert^2$. I'm ...
0
votes
2answers
25 views

Taylor series for $\frac{1}{az+b}$ centered at $z=0$ by substitution

I need to find the Taylor series centered at $z=0$ (i.e., the Maclaurin series) for $\displaystyle \frac{1}{az+b}$, where $a,b \in \mathbb{C}$ and $b \neq 0$. I thought it would be good to start out ...
2
votes
1answer
29 views

Branches of $\log(z^2)$

Suppose $z=x+iy$, then $r=\sqrt{x^2+y^2}$ and $\theta_p=\tan^{-1}\left(\frac y x\right)$, so we have $$z=re^{i\theta},\theta=\theta_p+2n\pi$$ Now, $z^2=x^2-y^2+i(2xy)$, so $r'=x^2+y^2$ and ...
-1
votes
0answers
29 views

square root negative reciprocal

I've been brushing up on some math and I stumbled upon a seeming contradiction with imaginary numbers. Based on these properties of imaginary numbers: $\sqrt{-1}=i$ $\frac{1}{i} = -i$ Then start ...
1
vote
3answers
33 views

Equality in generalized triangle inequality

The problem is to show that if $z_1,\dots,z_n$ are complex numbers then $$|z_1+\cdots+z_n|=\sum |z_i|$$ if and only if $$\mbox{arg}(z_i)\equiv \mbox{arg}(z_j)\mod 2\pi$$ for all $i,j$. I can ...
1
vote
0answers
37 views

Show that $\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad R \rightarrow \infty$

Given the straight line in the complex plane: $b+iR$ to $b+1+iR$ where $0<b<1$ and $|Im(a)|<\pi$, show the following: $$\int_{C} \frac{e^{az}}{\sin(\pi z)} \,dz \rightarrow 0 \quad as \quad ...
0
votes
1answer
21 views

For which values of $z \in \mathbb{C} $ do we have $ A_2 (z) = 0 $?

$ \forall z \in \mathbb{C} $ : $ e^{ jz} = \displaystyle \sum_{ n \geq 0 } \dfrac{ (jz)^{n} }{n!} = \displaystyle \sum_{ n \geq 0 } \dfrac{ x^{3n} }{ (3n)!} + j \displaystyle \sum_{ n \geq 0 } \dfrac{ ...
0
votes
1answer
24 views

Representing particular sets of complex numbers

I am supposed to represent the following sets: \begin{align}A&=\{z\in\mathbb{C}:\Re(2z+iz)<0<\Im(z^2)\}, \\ B&=\{w\in\mathbb{C}:w=z^2, z\in A\}, \\ C&=\{u\in\mathbb{C}:u=1/z, ...
0
votes
2answers
113 views

If $e^{i\pi}=-1$, then what does $e^{2i\pi}$ equal?

As the question says. As according to Euler's formula, $e^{i\pi}+1=0$ thus $e^{i\pi}=-1$, what therefore does $e^{2i\pi} $ equal?