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

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3
votes
4answers
206 views

Solution to this complex number equation

Solve $z^5 +32 =0$ My attempt : $$z^5 = -32$$ Multiply the powers on both sides by $\frac{1}{5}$ we get $$z = 2 * (-1)^\frac{1}{5}$$ Now I'm stuck at this step I don't know how to ...
1
vote
1answer
40 views

Write $(z-w)$ as $a+bi$, where $z = 8 - 4i$ and $w = 8 + 4i$

If $z = 8 - 4i$ and $w = 8 + 4i$, then write the expression $(z-w)$ in the standard form $a + bi$. However when I do that I get $0$, because from what I assume $8-4i$ and $8+4i$ cancel each other out....
1
vote
1answer
49 views

Five roots on an ellipse in the complex plane [closed]

What is special with an originating fifth order polynomial with one real and four complex roots lying on an ellipse when plotted as vectors in complex plane? If three in number in the special case ...
0
votes
0answers
29 views

Extension of Scalars of Complex Numbers

If we consider the complex numbers as a $\mathbb{R}$-module (vector space in this case), then its natural extension of scalars to $\mathbb{C}$ seems to be the complex numbers themselves, with the ...
7
votes
2answers
178 views

Compute complex integral resulting from FT

I obtain the following integral after doing a FT of a function $$\int_{-\infty}^{\infty} e^{-\pi(x + i\xi)^2}dx$$ I am not sure how to evaluate it. I tried change of variable $y = x+ i\xi$. but what ...
0
votes
1answer
72 views

Finding the centre of the circle on the complex plane

The centre of the circle represented by $|z+1|$=$2|z-1|$ on the complex plane is (a)0 (b)5/3 (c)1/3 (d)None of these What I've tried so far in attached in the pic below. Please refer to it
1
vote
1answer
26 views

Sketching Complex region

This is the question: Conisder the points in the region $R$ shown in the Argand diagram of Figure 2, consisting of all points in a right-angled sector of radius $1$, except for the point $ z ...
0
votes
2answers
80 views

Solve $z^5=-32$ and draw its solutions in complex space, then describe their characteristic geometrical property.

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...
1
vote
1answer
39 views

A fallacy in the imaginary numbers. [duplicate]

$$\sqrt{-5}*\sqrt{-3}=\sqrt{-1*5}*\sqrt{-1*3}$$ $$\sqrt{-1*-1}*\sqrt{5*3}=\sqrt{5*3}$$ $$=\sqrt{15}$$ But we all know that this below is right, $$\sqrt{5}i*\sqrt{3}i=-\sqrt{15}$$ So, please explain ...
1
vote
5answers
111 views

Difference between real and complex solutions of cubic equations [closed]

Take for an example, this equation. $$x^3+15x+4=0$$ This equation has two complex solutions and a real one. $$x≈0.1327-3.8798i$$ $$x≈0.1327+3.8798i$$ $$x≈-0.26542$$ What's extra in the complex ...
0
votes
3answers
54 views

Finding the number of roots of the given equation

Number of roots of the equation $z^{10}-z^5-992=0$ where real parts are negative is (a) 3 (b)4 (c)5 (d)6 What I've tried so far Let $z=x+iy$ Now, putting the value of $z$ in the equation, we ...
0
votes
1answer
61 views

Computing Complex Line Integrals

I'm having trouble understanding exactly how to compute a complex line integral in $\mathbb{C}$. With my understanding of multivariable calculus, I view the line integral of a vector field $F: \mathbb{...
12
votes
1answer
1k views

Does an iterated exponential $z^{z^{z^{…}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $...
0
votes
0answers
16 views

How do I determine and sketch the images $g(\mathbb{R})^2$ as sets and as geometric objects?

It's given function $g(x, y) = \begin{pmatrix}e^x \cos y\\ e^x \sin y\end{pmatrix}$. How do I determine and sketch the images $g(\mathbb{R})^2$ as sets and as geometric objects?
0
votes
1answer
33 views

Find a linear map $f_{\theta}: \mathbb R^2 \to \mathbb R^2$ which describes rotation by $\theta$ in counterclockwise direction

$\theta \in [0, 2\pi).$ Hint: for a given angle $\theta$, find $a, b, c, d \in \mathbb R$ such that $f_{\theta}(x_1, x_2) = (ax_1 + bx_2, cx_1 + dx_2).$ This problem occurs at the end of a ...
1
vote
2answers
43 views

Equilibrium in a system of nonlinear differential equations

I have two questions about a specific system of differential equations. First, if a complex number can be an equilibrium point. Second, and related with the first question, how can I verify that $(0,0)...
0
votes
0answers
31 views

holomorphic square root

Let $A_{R,r}=\{z\in\mathbb C: r\lt|z|<R\}$ Prove that there can not be a function $q \in O(A_{R,r})$ such that $q^2(z)=z$ $z=a*e^{\phi *i}, \; r\lt a \lt R, \phi \in [0;2\pi]$ and the square ...
0
votes
1answer
16 views

Finding the equation of the locus in Cartesian form

Sketch the locus determined by arg(z+2+i)=-pi/6 Find the Cartesian equation of this locus I have no idea how to do this, but i have an idea that the locus will be a circle
1
vote
1answer
20 views

If $g(x) = 1-2s(1−\cos(x))−ic \sin(x)$, show that $|g(x)| \le 1$ iff $0\le c^2 \le 2s \le 1$.

Question: If $$g(x) = 1-2s(1−\cos(x))−ic \sin(x),$$ show that $|g(x)| \le 1$ iff $0\le c^2 \le 2s \le 1$. My attempt: Ugh I am completely stuck, here. I've tried finding $|g(x)|^2$, which is: $$|g(...
0
votes
1answer
24 views

Consider the complex numbers z1=-8√2-(8√2)i and z2=2√3+2i

a) show clearly that 2cis(13π/16) is a fourth root of z1. I was able to answer this by showing that 2cis(13π/16) to the fourth power is equal to z1. b) find the other fourth roots of z1 and plot ...
5
votes
8answers
143 views

Find $\sqrt{8+6i}$ in the form of $a+bi$

I need help with changing $\sqrt{8+6i}$ into complex number standard form. I know the basics of complex number such as the value of $i$ and $i^2$, equality of complex number, conjugate and ...
0
votes
1answer
30 views

The function $\zeta(\frac{1}{2}+it) \left[ \sqrt{2}\left( \cos(t\log 2)+i\sin(t\log 2) \right) -2 \right]$ has a numerical root

Using the complex exponentiation (this is the MathWolrd's Page) one can deduce for $t>0$ $$2^{\frac{1}{2}+it}=\sqrt{2}e^0(\cos(t\log 2)+i\sin(t\log 2)),$$ since $a=2,b=0,c=\frac{1}{2}, d=t$ and $\...
-3
votes
1answer
52 views

How do you express cos(nx) + sin(nx) in terms of eulers constant [closed]

The examples I have seen express $\cos(nx) + i\sin(nx)$ which comes to $e^{i\,nx}$. Is there a way to use Euler constant when both the cos and sin have the same coefficients?
0
votes
0answers
11 views

Show that if $\xi_1 = z + \sqrt{z^2 +1}$, and $\xi_2 = z - \sqrt{z^2 + 1}$, then $|\xi_{1,2}| \le 1$ or, if $\xi_1 = \xi_2$, $|\xi| < 1$

Question: Show that if $\xi_1 = z + \sqrt{z^2 +1}$, and $\xi_2 = z - \sqrt{z^2 + 1}$, then $\xi_1$ and $\xi_2$ satisfy the conditions: $|\xi_j| \le 1$, for $j = 1,2$, and if $\xi_1 = \xi_2$, then $|\...
3
votes
1answer
32 views

Trigonometric functions and complex numbers

I solving the inverse Laplace transform using the method of Heaviside. This is part of the problem: I understand the division between complex numbers and that $e^{it} = Cos(t) + iSin(t)$, but I ...
1
vote
1answer
22 views

Multiplying complex vector by a complex number

How can I geometrically interpret multiplying a complex vector by a complex number. Let's say I have a vector $(a, b)$ in a complex vector space. Now, let's say I multiply this vector by $re^{i\theta}$...
1
vote
1answer
35 views

$|1-e^{2\pi i n z}|<1$ for $z$ in the upper half plane

If $z$ is in the upper half plane, then $|e^{2 \pi ni z}|<1$ for every $n\in\mathbb{N}$. But why is also $|1-e^{2\pi i n z}|<1$? I just get $$|1-e^{2\pi i n z}|\leq1+|e^{2 \pi ni z}|<1+1=2....
1
vote
0answers
13 views

The multivalued behaviour of complex exponential $z^\lambda$

On Gustav Doetch's Introduction to the Theory and Application of the Laplace Transform, it says: The power series $\sum_{n=0}^\infty a_nz^n$ converges on a circular disc. Replacing the integers $n$...
2
votes
0answers
45 views

complex number quotient and proof

I am working on a complex number problem set and I have had success so far except for the following, which makes me think that I am missing something rather simple in the initial steps that would ...
1
vote
1answer
40 views

Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$

I have seen this identity on Wolfram mathworld and in a comment to another similar trigonometric proof: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I can't seem to find a ...
0
votes
1answer
34 views

Simple Complex calculus

Suppose i wanna solve $e^{iz}=-1-\sqrt{2}$ now that is $$iz=ln|-1-\sqrt{2}| +i(Arg(-1-\sqrt{2})+2kπ)$$ since $-1-\sqrt{2}$ is real $Arg=0$ so $z=-iln(1+\sqrt{2})+2kπ$ $k$ is an integer. Am i wrong? ...
0
votes
0answers
40 views

Is $-4^2 = -8i$?

Hey sorry I'm new to this topic, but here is what got me there. After I found out, that $\sqrt{2i} = i+1$, I realised that it makes perfect sense if you visualize it on the plane with triangles. So ...
3
votes
2answers
51 views

What are the values of $a_0,a_1,…,a_{10}$ if $\cos^{10} {\theta}=\displaystyle\sum_{k=0}^{10}a_k\cos {k\theta }$?

I was thinking of doing the following. Let $A=a_o+a_1\cos {\theta }+a_2\cos {2\theta }+...+a_{10}\cos {10\theta }$ and $B=a_1\sin {\theta }+a_2\sin {2\theta }+...+a_{10}\sin {10\theta }$ Then, ...
0
votes
2answers
25 views

Imaginary Number Inequalities

I've been thinking about lately complex numbers and inequalities, and after establishing there is no way to order all complex numbers I thought can you order imaginary numbers? I started by ...
0
votes
1answer
25 views

Convergence of roots in the unit disk of $\mathbb C$

Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers such that $z_n^n=z$ for each $n\in\mathbb N$, where $z\in\mathbb C$ is a fixed complex number on the closed unit disk: $|z|\leq 1$. I want ...
2
votes
2answers
58 views

Show that $2\cosh z + \sinh z = i$

The equation is $$2\cosh z + \sinh z = i$$ I used the following formulas: $$\cosh z = \frac{e^z + e^{-z}}{2}, \sinh z = \frac{e^z - e^{-z}}{2}$$ to reduce this equation to $$3e^z - e^{-z} = 2i$$ but ...
1
vote
1answer
22 views

Complex numbers inequalities and optimisation

I'm now aware that you can't definitely with ease say that one complex number is greater than another. Though what about imaginary numbers? Is $5i > 3i$? Is $i>-i$? Is it possible to optimise (...
5
votes
4answers
470 views

Can we determine if a complex number is greater than another? [duplicate]

Is it possible to determine if one complex number is greater than another? Or as the question implies is there an "order" to complex numbers (like 1 is before 2 in the real numbers)? I thought that ...
1
vote
1answer
20 views

graphing multiple functions limited between between two x values

On a graph you can limit the showing of a function between two x values by multiplying by a complex when outside. $$\\f(x) = ax * \frac{(\sqrt{(max -x)} * \sqrt{(x - min)}}{(\sqrt{|(max -x)|} * \sqrt{...
1
vote
1answer
45 views

Simplify a polynomial in $e^{i\omega}$ without using trigonometry

Given $z=e^{\omega i}$, the following polynomial can be reduced to a quadratic using trigonometry. $$ P_0 (z^3 - z^{-3}) + P_1 (z^2 - z^{-2}) + P_2 (z^1 - z^{-1}) $$ One method is to exploit the ...
0
votes
1answer
63 views

Why is $i^i=\exp(-\pi/2)$, where $i$ is the imaginary unit.

I looked up $i^i$ and it is said that it is equal to $\exp(-\pi/2)\approx 0.20787$. But I tried the following: $$i^i=\exp[\ln{(i^i)}]=\exp[i\ln(i)]=\exp(i\ln(e^{i(\pi/2+2\pi k)}))$$ $$=\exp(i\cdot i(\...
1
vote
2answers
49 views

Notation $e^{\pm \pi \cdot i}$

I am confronted with the equations as $(1-t)=(t-1)e^{\mp\pi i}$, or $(1-e^z)=e^z(1-e^{-z})e^{\mp\pi i}$ and alike. I am wondering what this means. Isn't it true that $e^{\pi i}= -1 =e^{-\pi i}$? Best ...
2
votes
2answers
95 views

Analytic floor function, why this seems to work?

I have been using this formula which I determined for myself for quite some time now for use in everything from the sgn() function to the Kronecker delta to the ceiling and NINT() functions but haven'...
-1
votes
0answers
24 views

How do I get the limit of exponential functions with complex number?

With WolframAlpha's help, I got, $$ \lim_{\tau\to-\infty}{e^{(1-i\omega)\tau} }= 0 $$ But, how did this work in complex number context?
3
votes
0answers
55 views

All solutions of $ z^i = i^z $

In the simple equation $ z^i = i^z $ how are all complex values found? $ z= \pm \, i, $ and what else? It can be found by inspection, but to find general solution: We take logs, there is a ...
1
vote
1answer
35 views

Circumference in a complex plane

In this question I learnt that the circumference of a unit circle in a complex plane is just the circumference of any normal circle $2\pi r$. Now I would like to take into account the imaginary ...
3
votes
1answer
65 views

What is the circumference of a complex unit circle?

What is the circumference of a unit circle in a complex coordinate system $ e^{i \phi} $ ?
-1
votes
1answer
30 views

Convert complex number to trigonometric form [closed]

I don't know how to change complex number $z=5+7i$ to trigonometric form
3
votes
1answer
106 views

What should $\int \frac{1}{x} dx$ equal to?

Before you say that $\int \frac{1}{x} dx$ is equal to $\ln|x| +C$ due to positve and negative, I would like to show you why it is not convincing to me. Problem 1 and its possible solution. \begin{...
2
votes
1answer
75 views

On a second-order differential inequality involving the Dirichlet eta function

After that I've tried understand the problem 6416 [1983, 60] A Second-Order Differential Inequality proposed by Sandford S. Miller in the American Mathematical Monthly (myself proposal is ...