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

learn more… | top users | synonyms

2
votes
3answers
41 views

Find all values of $\sqrt[4]{-1+i}$

Okay. I know how to solve for all values of $\sqrt{-1} $ but $\sqrt{-1+\iota} $ confuses me a bit. I got the value of r to be $\sqrt 2 $ I ended up with this: $ z_k = ...
0
votes
4answers
59 views

is 1 greater than i?

I'm not sure this question even makes sense because complex numbers are a plane instead of a line. The magnitudes are obviously the same because i is a unit vector, but is there any inequality you can ...
1
vote
2answers
22 views

$Im(\cosh z)=\sinh x\sinh y$ and $|\sinh z|^2=\sinh^2(z)+\cosh^2(z)$

I have problems in two issues of complex variables ... 1) Prove that $Im(\cosh z)=\sinh x\cosh y$, if $z=x+iy$. I tried to expand $\cosh z= ...
2
votes
1answer
45 views

Prove that $f$ has a simple pole at $z=0$

Let, $f:\{z\in \mathbb C:0<|z|<1\}\to \mathbb C$ be analytic such that $n\le |f(1/n)|\le n^{3/2}$ for $n=2,3,...$. Assume that $z^2f(z)$ is bounded in $|z|<1$. Show that $f$ has a pole of ...
0
votes
6answers
63 views

Find the three roots of $z^3 = -i$ in the form $a+ib$. [on hold]

Find the three roots of: $$z^3 = -i $$ in the form $a+ib$.
1
vote
0answers
24 views

Locus of complex numbers.

Question Let $P(x,y)$ be the point on an Argand diagram representing the complex number $u=x+iy$ and satisfying the equation \begin{align*} \vert u \vert=k\vert u+a\vert, \end{align*} where $k$ is ...
0
votes
1answer
35 views

$|z^i|<e^\pi,\;\;\forall z\in\mathbb{C}-\{0\}$

Good morning people ... Do you have any idea to help me prove that $$|z^i|<e^{\pi}$$ for $z\in\mathbb{C}-\{0\}$. I tried to do $z^i=e^{i\ln z}=e^{i(\ln r+i\theta+2k\pi i)}$ if $z=re^{i\theta}$, ...
2
votes
0answers
35 views

Are there famous complex constants?

Are there any famous constants (like $\pi$ and $e$) that are complex? More specifically, to rule out trivial complex numbers, are there any famous constants of the form $a+bi$ with $a \neq 0 \wedge b ...
0
votes
1answer
16 views

Rotations of complex graphs

Let $c_1 = -i$ and $c_2 = 3$. Let $z_0$ be an arbitrary complex number. We rotate $z_0$ around $c_1$ by $\pi/4$ counter-clockwise to get $z_1$. We then rotate $z_1$ around $c_2$ by $\pi/4$ ...
1
vote
1answer
24 views

Rotation in the complex plane

The function $f(z) = \frac{(-1 + i \sqrt{3}) z + (-2 \sqrt{3} - 18i)}{2}$ represents a rotation around some complex number $c$. Find $c$. Hello, I am having some trouble trying to do this problem. ...
-1
votes
1answer
17 views

Cartesian equation of the loci of 'z', [on hold]

Can someone FIND |z-3j|+|z+7j|=12
1
vote
1answer
30 views

Consider the equation $|z + 3i|=3|z|$ for complex z and give a geometric description of the set S of all solutions.

Writing $z$ in the form $a+ib$ and then rearranging gives $-8a^2-8b^2+6b+9=0$. The most promising form I could manage from this is $(b-\frac{3}{8})^2=(\frac{9}{8}-a)(\frac{9}{8}+a)$ but I still do not ...
-9
votes
1answer
52 views

Can someone find z^10 quickly, given z? [on hold]

z = 1 + i z^10 = ? I need a solve of this quick please !
0
votes
1answer
28 views

How can I split this into its' real and imaginary parts, and simplify?

Essentially, I want to prove that $| \sum_{k=1}^n e^{ik}|$ is bounded. If I obtain an expression for this sum: $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ I am not sure how to proceed ...
0
votes
1answer
40 views

Help with simplifying this - where have I made an error

$$e^i \frac{e^{in}-1}{e^i-1}$$ $$e^i \frac{e^{in}-1}{e^i-1} \cdot \frac{e^{-i} -1}{e^{-i} -1}$$ $$e^i \frac{(e^{in}-1)(e^{-i}-1)}{(e^{-i}-1)(e^i-1)}$$ $$e^i ...
3
votes
5answers
42 views

Find the root of a complex number

Find all complex numbers $z$ such that $$z^2=12−16i,$$ and give your answer in the form $a+bi$. We set $$z= a+bi,$$ thus, $$z^2 = (a^2 - b^2) + (2ab)i.$$ Equating both $z^2$ we have $$ a^2 ...
2
votes
1answer
29 views

How to solve a Complex equation involving the conjugate: $(x+iy)^2-2(x-iy)+1=0$

I want to find a Complex value for $z$ that satisfy the equation: $$z^2-2z^*+1=0$$ But i have never seen the conjugate taking part of an equation. What i have tried is give $z$ some components ...
0
votes
2answers
22 views

How do you compute an expression containing complex numbers with large powers?

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$ $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$ So I have these equations with the answers ...
0
votes
1answer
27 views

Solve z in an expression involving complex conjugates.

Solve for z, and give your answer in the form a+bi. $$\overline{z+2-2i} = {2z + 5 - 7i}$$ I know fully understand the concept of complex numbers and complex conjugates. I've found that the answer is ...
0
votes
2answers
43 views

Understanding complex numbers

I need to show that $$\left | \sum_{k=1}^n e^{ik}\right | $$ is bounded Now I am given that $$ \sum_{k=1}^n e^{ik} = e^i \frac{e^{in}-1}{e^i-1}$$ But have little idea of how to proceed further and ...
0
votes
3answers
44 views

Roots of Unity, Precalculus

(a) Let $\omega$ be a complex number such that $\omega^5 = 1$ and $\omega \neq 1$. Find $$\frac{\omega}{1 + \omega^2} + \frac{\omega^2}{1 + \omega^4} + \frac{\omega^3}{1 + \omega} + \frac{\omega^4}{1 ...
1
vote
0answers
26 views

Prove that if $g = r +ip$ is analytic on $C$ and $r(x,y) \leq M$, with $M > 0$, for all $(x,y)\in C$, $g$ is constant.

Let $g = r +ip$ be analytic on $C$. If for some $M > 0$ we have $r(x,y) \leq M$ for all of $C$, then $g$ is constant. The theorem is given without proof in my notes and I can't find any examples ...
0
votes
1answer
20 views

Shade on your Argand diagram the region $\frac{\pi}{4}\,{\le}\,\arg\,z\,\le\frac{\pi}{2}$

Is this saying the region from $\arg\,z=\frac{\pi}{4}$ to $\arg\,z=\frac{\pi}{2}$ in an anticlockwise direction? How would you represent the region from $\arg\,z=\frac{\pi}{4}$ to ...
-2
votes
0answers
23 views

Locally Lipschitz complex function [on hold]

I want to study the property of being locally Lipschitz for the following function $$f(z)=\vert z\vert^\gamma z^2$$ with $\gamma\in\mathbb{R}$. Some hints to study this problem?
1
vote
1answer
28 views
+50

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + ...
-4
votes
0answers
33 views

How to integrate complex numbers? [on hold]

Complex numbers have 2 variable so does it's integration entail contour integration or can we integrate assuming one variable to be a constant in terms of the other or do we try to find a relation ...
-4
votes
0answers
34 views

Complex Analysis Exam tomorrow, what are some good to know facts? [on hold]

The course covers differentiation, integration, series, and a lot of theorems. What would you say is crucial to know for an exam of undergrad complex analysis?
2
votes
4answers
55 views

Showing that linear transformations $1, T, T^2, T^3 ,\dots $ do not span the set of linear transformations of $ \mathbb C^n$ into $ \mathbb C^n$

Suppose $T : \mathbb C^n \rightarrow \mathbb C^n$, $n \geq 2$ is a linear transformation. Show that the linear transformations $1,T, T^2, \dots$ do NOT span $L(\mathbb C^n, \mathbb C^n)$, the set of ...
0
votes
2answers
30 views

Trisecting a line in the complex plane

We have $x = 11-13i$ and $y = 35-i$. $a$ is a complex number which trisects the line segment joining $x$ and $y$. $a$ is also closer to $x$ than $y$. Find $a$. I'm not sure where to start. Would a ...
0
votes
0answers
20 views

Linear interpolation of complex numbers

Does it make sense to say that, if I have two numbers $X = a + bi$ and $Y = c + di$, I can approximate a point between them as $Z = \frac{a + c}2 + \frac{b + d}2i$, interpolating the real and complex ...
0
votes
2answers
38 views

Minimising $|a+bw+cw^2|$ such that a,b,c are consecutive integers?

Suppose we are given a expression $k=|a+bw+cw^2|$ such that $w$ is cube root of unity ($w\neq1$) such that $\{a,b,c\}$ are consecutive integers , then how can we minimise value of expression ? I was ...
1
vote
1answer
21 views

Integral using complex numbers shortcut

I want to compute the following integral $$- \frac{1}{M(\lambda_1-\lambda_2)}\int\limits_{-\infty}^t(e^{\lambda_1(t-t')}-e^{\lambda_2(t-t')})(\beta\omega A\sin\omega t' +g)\;dt'$$ Here the integral ...
19
votes
8answers
4k views

Why did Euler use e to represent complex numbers?

From Euler we've learned that $z=re^{i\theta}$. And it's easy to see that $|z|^2=r^2$, since $re^{i\theta}\times re^{-i\theta}=r^2$. Why must we use e to represent these numbers correctly? It seems ...
0
votes
0answers
24 views

Complex expansion of this matrix

If $$M_{IJ}$$ is a symmetric complex matrix, is this expanded as a usual complex function, that is to say $$M_{IJ}=Re(M_{IJ})+ iIm(M_{IJ})$$ or for some different reason, this can be expanded as ...
3
votes
2answers
71 views

Can a number have both a periodic an a non-periodic representation in a non-integer base?

Fix an algebraic number $\beta$ and consider a complex number $\alpha$ which admits multiple representations in base $\beta$. If one representation of $\alpha$ is ultimately periodic, must every other ...
0
votes
1answer
33 views

How to graph $|z-1| <2$

Am I correct to rearrange this to $(z-1)^2 < 4$, and hence just graph as a circle or am I completely off?
3
votes
2answers
102 views

Complex number weird sum.

Question given in red. My working in black. $$\color{red}{\sum_{r=0}^{50}z^r=0}\iff z_k=\exp\underbrace{\left(\frac{{\cal i}2\pi k}{51}\right)}_{\theta_k},k\in\{n\mid n\le50,n\in\mathbb ...
-1
votes
0answers
69 views

Is it possible to take infinite sums of infinite sums?

I learned a few weeks ago that any diverging series can be given a specific value, not its limit ( for there are no limits in a diverging series), rather, "an average value" of the sum. From so, I ...
0
votes
1answer
34 views

Rotation group of $20$ degrees

Let $R_{20}$ be a rotation counterclockwise by $20$ degrees in the $xy$ plane. What is this group? Then re-write the group in terms of complex numbers of the form $e^{i\phi}$. Is their a special ...
0
votes
2answers
34 views

Calculate $\int_\Gamma ze^{z}dz$ where $\Gamma$ is line from point $z_1=0$ to point $z_2=\frac{\pi i}{2}$

$$ \int_\Gamma ze^{z}dz\ $$ where $\Gamma$ is line from point $z_1=0$ to point $z_2=\frac{\pi i}{2}$ Hello, pls. how correctly calculate this example? I don't know what do with exponent..
0
votes
2answers
149 views

Proving $x^{2}+1 \neq n! $,using Gaussian Integer.

I want to show that $$x^{2}+1 \neq n! $$ for $n>3$ where $x,n$ are both integers. Since $$x^{2}+1=(x+i)(x-i) $$ it follows that $x^{2}+1$ has only prime factors on the form ($4k+1$), whereas $n!$ ...
0
votes
2answers
53 views

How should this equation be read, $|z+1+3i|=|z-5-7i|$

$$|z+1+3i|=|z-5-7i|$$ $z$ represents a complex number right? Then if $$|z+1+3i|=0$$ $${\implies}|z|=|-1-3i|$$ In which sense does this $$|z+1+3i|=|z-5-7i|$$ imply, $$\implies|-4-4i|$$ But $z$ has ...
0
votes
1answer
15 views

Calculate the maximum value of the complex modulous |v-u|

I was asked to sketch the complex where $|u-1+i| = 2$ and $|v-(3+5i)|=1$ on the same diagram then I was asked to find the maximum value of |v-u| What is $|v-u|$ basically ? how am I going to ...
1
vote
2answers
24 views

Why does the function $f(z) = 1/\sin(\pi/z)$ have isolated singular points?

In the complex analysis text book "Complex Variables and Applications 8th edition", it states the function $1/sin (\pi/z)$ has singular points $z = 0$ and $z = 1/m; (m = \pm 1,2,3,4,\dots.) $ I sort ...
0
votes
0answers
16 views

Switching between Cartesian coordinate and polar coordinates

Under what assumption, every non-zero complex number represented in Cartesian coordinate system admits unique polar representation and vice versa ?
1
vote
2answers
35 views

When the argument of complex numbers is a well defined real valued function?

I know that the argument $\arg:\Bbb C\setminus\{0\}\to\Bbb R$ is multivalued function and also that if we consider $\arg:\Bbb C\setminus\{0\}\to{\Bbb R}/{2\pi \Bbb Z}$, then it is a well defined ...
3
votes
2answers
64 views

Reference to complete proof that integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$?

Where can I find a complete proof to the fact that the integral closure of $\mathbb{Z}$ in $\mathbb{Q}(i)$ is $\mathbb{Z}[i]$ (the Gaussian integers are the integral closure of $\mathbb{Z}$ in the ...
0
votes
1answer
51 views

argument of $z = \sqrt 5 + 5i$?

I know that because both $a$ and $b$ are positive it is in the first quadrant and hence $\arg z$ should just equal to $\arctan(b/a)$, but I've been told that the answer is $\arg z= \arctan \sqrt5 $??? ...
0
votes
2answers
30 views

Why is $(-1)^p$, where, $0<p<1$ is a complex number?

Why is $(-1)^p$, where, $0<p<1$ is a complex number? If $p = \frac{1}{4}$, then $(-1)^{p}=(-1)^\frac{1}{4}=((-1)^{4})^\frac{1}{16}=(1)^{1/16}=1$ However, apparently $(-1)^\frac{1}{4}=a+ai$, ...
0
votes
0answers
28 views

Is every natural recursive relation necessarily holomorphic?

Define the set of algebraic primitive recursive relations as the set of functions defined by: $$ F(n,a,k) = F(n-1,F(n-1,F(n-1,a,a),a)...,a)_{\text{nested to depth k}}$$ $$ F(0,a,k) = a + k $$ Along ...