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

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3
votes
4answers
43 views

Problem involving complex conjugate

I have the equation: $$3z-\bar{z}=2-3i$$ First I write this as: $$3(x+yi)-(x-yi) = 2-3i$$ $$3x-3yi-x+yi = 2-3i$$ $$2x+4yi = 2-3i$$ Now the following must be true: $$2x = 2\quad\mbox{and}\quad 4y ...
0
votes
0answers
21 views

Stuck on A level Loci question

The transformation at $T$ given by $w=kz/(i+z)$ where $z\neq -i$, $k$ a real number, maps the complex number $2+i$ in the $z$-plane to its image $1/2(3-i)$ in the $w$-plane. a) Show that $k=2$ Point ...
0
votes
1answer
27 views

When are the functions $x_1=Ae^{iωt} + Be^{-iωt}$ and $x_2=Ae^{-iωt} + Be^{iωt}$ identically equal?

Suppose we have two equation $x_1=Ae^{iωt} + Be^{-iωt}$ and $x_2=Ae^{-iωt} + Be^{iωt}$. Where $A$ and $B$ are complex number and $A^*$ and $B^*$ are their conjugate correspondingly. Now if we want to ...
49
votes
15answers
4k views

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
4
votes
0answers
54 views

Complex numbers and geometry - Four complex numbers lying on a circle

I'm stuck on a problem of What is Mathematics?, by Courant and Robbins. The formulation is as follows: Prove that if for four complex numbers $z_1$, $z_2$, $z_3$ and $z_4$, the angles of ...
0
votes
2answers
43 views

Determine the image of the strip $S$ consisting of all points $z$ with $\frac{-\pi}{2}\lt Re(z) \lt \frac{\pi}{2}$ and $Im(z)>0$ under $w=i\sin z$

$\color{green}{\text{transformation is}\space w=i\sin z}$ $$w=i\sin z = i\sin(x+iy)=\frac{1}{2}\left(e^{ix-y}-e^{-(ix-y)}\right)=-\cos(x)\sinh(y)+i\sin(x)\cosh(y)$$ $\therefore u = -\cos(x)\sinh(y) ...
3
votes
3answers
113 views

Can $x^n = 1$ describe the equation of a unit circle?

The complex roots of the equation $x^n = 1$ lie on the unit circle. Suppose $n$ goes large. Is it correct to say that all the roots of $x^n=1$ form a circle of radius one?
0
votes
2answers
44 views

Find the region in the w-plane to which the line y = 1 is transformed by $\frac{1}{z}$

I tried to do the following: $$w=\frac{1}{z}=\frac{x-iy}{x^2+y^2}$$ $\implies u = \frac{x}{x^2+y^2} and\space v = \frac{-y}{x^2+y^2}$ $\color{green}{need\space to\space transform\space the\space ...
0
votes
1answer
24 views

Showing that $f(z)$ is differentiable throughout a region

I'm having trouble with an algebraic operation in a proof, which I will copy here: Specifically, I do not see the connection between steps 4.8 and 4.9. As best as I can tell, the equations in 4.8 ...
4
votes
2answers
233 views

Is this derivation of $(5i+9)(5i−9) = -106$ correct?

I was simplifying this problem for a class exercise the other day that looked something like this: $$(5x+9)(5x-9)$$ Obviously the simplified version of this is $25x^2-81$, but I wondered to myself, ...
0
votes
1answer
44 views

Modulus of exponential function with real and complex arguments

Can anyone please explain why $$|e^{\frac12 \sin(2x) }|\le e^{1/2}$$ for all real $x$, while $$|e^{-i\sin(x)^{2}}|=1$$ for all real $x$?
1
vote
2answers
38 views

How to find the complex roots of $x^2-2ax+a^2+b^2$?

How to find the complex roots of $x^2-2ax+a^2+b^2$? I tried using the quadratic formula: $$ x_{1,2} = \frac{2a \pm \sqrt {4a^2-4b^2}}{2} = {a \pm \sqrt {a^2-b^2}} = a\pm \sqrt{a-b}\sqrt{a+b}$$ ...
3
votes
1answer
315 views

Complex number $\ln(i)$ into rectangular form?

I am going through the Mathematics of the DFT book by Julius O. Smith III. One of the questions ask the following: How would you convert the complex number $\ln(i)$ into the form $x + yi$, ...
1
vote
3answers
643 views

Defining the Complex numbers

I posted this question nearly 10 days ago, but am still really not satisfied with the answers I got, I have no prior education in abstract algebra, group theory, or other abstractions, and most of the ...
3
votes
4answers
70 views

How does $\dim \mathbb C$ work?

In the Wikipedia page about Dimension (vector space), it says the dimension of complex numbers is 2 or 1 if it's complex or real vector space respectively. How does that work? How to I describe ...
2
votes
1answer
66 views

Can someone please explain the following definition of $\ln(e^z)$

I noticed someone do this from one of the questions is asked on here i had: $$e^z = -0.5$$ $$e^z = 0.5e^{i\pi}$$ which magically became: $$z = \ln\left(\frac12\right) + iπ + 2ikπ$$ does this mean ...
1
vote
1answer
49 views

Alternative Solution to a complex numbers problem

Let $z \in \mathbb C$, such that $z = x+ix, \; \forall x \in \mathbb R^* $ Prove that $$K(z) = \frac {z^4 + z^8 + \cdots+ z^{4n}} {iz^2 + i^5z^6 + \cdots+i^{4n-3}z^{4n-2} } = \mathrm {Im} ...
0
votes
1answer
62 views

Is he using a theorem for real numbers, on a complex power series?

This is from Rudins principles of mathematical analysis. First are theorems 3.41 and 3.42 which he uses later. I assume that 3.41 holds for complex numbers? But what about 3.42?, complex numbers ...
12
votes
8answers
18k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
5
votes
1answer
214 views

Show $f$ is analytic if $f^8$ is analytic

This is from Gamelin's book on Complex Analysis. Problem: Show that if $f(z)$ is continuous on a domain $D$, and if $f(z)^8$ is analytic on $D$, then $f(z)$ is analytic on $D$. (I assume the ...
1
vote
1answer
48 views

Using the properties of real numbers, verify that complex numbers are associative and there exists an additive inverse

I am self-learning, so I need guidance, as I am unsure whether my approach is sufficient. There are two questions, both asking to verify a property of the complex numbers using the properties of real ...
2
votes
1answer
50 views

Show complex sequence is convergent

We have complex sequence $a_n$ such that $\displaystyle \sum_{n=1}^{\infty}a_n$ is convergent. Let $\sigma : \mathbb{N} \to \mathbb{N}$ be a bijection where we know that there exist $M \in \mathbb{N}$ ...
2
votes
2answers
341 views

2x2 inverse of a complex matrix with complex determinant

Firstly, my question may be related to a similar question here: Are complex determinants for matrices possible and if so, how can they be interpreted? I am using: $$ \left(\begin{array}{cc} a&b\\ ...
8
votes
2answers
113 views

In a complex vector space, $\langle Tx,x \rangle=0 \implies T = 0$

Suppose $T$ is a linear operator on a complex inner product space. Is it a theorem that if $\langle Tx,x\rangle=0$ for all $x$ in the space then $T=0$. The theorem fails in the real case, as seen for ...
0
votes
1answer
38 views

Why from $e^{i\mu}-e^{-i\mu}=0$ we can conclude that $e^{2i\mu}=1$?

Could you please explain why from $e^{i\mu}-e^{-i\mu}=0$ we can conclude that $e^{2i\mu}=1$ and $2i\mu=2n\pi i$, when $\mu$ is real or complex? I tried to use Euler's formula but without any ...
3
votes
2answers
66 views

if $f(z),\overline {f(z)}$ are analytic then they are constant

I'm trying to prove this "theorem": if $f(z),\overline {f(z)}$ are analytic in some open set $\Omega \subseteq \mathbb C$, then $f(z)$ is a constant. Hint: Use Cauchy-Riemann equations to show that ...
4
votes
4answers
309 views

Complex power of a complex number

Can someone explain to me, step by step, how to calculate all infinite values of, say, $(1+i)^{3+4i}$? I know how to calculate the principal value, but not how to get all infinite values...and I'm ...
0
votes
2answers
24 views

Locus of complex number in complex plane

I have the following complex number: $G = \xi + i\eta$ $\xi = 1-\sigma(1-\cos\phi_m)$ $\eta = -\sigma\sin\phi_m$ how can I find the locus of this complex number? I am told without proof that it is ...
1
vote
0answers
41 views

proof that the expression is Real for any $z$

Please help me with this problem, I'm clueless here. $\ \ \ \ (\bar{z}+1-2i)^{1985} + (\bar{z}+1+2i)^{1985}$ $\ \ \ \ $proof that the expression is Real for any $z$
0
votes
1answer
54 views

plotting $\frac{-\pi}{2}<x<\frac{\pi}{2} $ and $ 0<y<1$ under mapping $w=\sin(z)$

i need to plot this $\frac{-\pi}{2}<x<\frac{\pi}{2} $ and $ 0<y<1$ under $w=\sin(z)$ mapping so what i did is $ y=0 , \frac{-\pi}{2}<x<\frac{\pi}{2} => -1<u<1 , v=0 $ $ y=1 ...
4
votes
3answers
169 views

Find solution of equation $(z+1)^5=z^5$ [duplicate]

I attempt to solve the equation $(z+1)^5=z^5$. My first approach is to expand the left hand side but ı get more complicated equation. So I couldn't go further. Secondly, I write equation as, since ...
1
vote
1answer
56 views

Find $(1+i)^i$ in simpler terms, without imaginary exponents. [duplicate]

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent. since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
2
votes
5answers
712 views

How does $e^{i x}$ produce rotation around the imaginary unit circle?

Euler’s formula states that $e^{i x} = \cos(x) + i \sin(x)$. I can see from the MacLaurin Expansion that this is indeed true; however, I don’t intuitively understand how raising $e$ to the power of ...
1
vote
2answers
165 views

How one can implement the equation with “$i$” in it?

I have an equation: $$f(t)=c(e^{i2\pi\frac{n}{T}t}+e^{-i2\pi\frac{n}{T}t})$$ ...for $t\in(-\pi,\pi)$, and with $T=2\pi$. I have to draw a plot of the function $f(t)$ for $n\in\left \{0,1,2,5 \right ...
0
votes
1answer
18 views

Superfunctions with complex iteration indices: Interpretation

Superfunctions are a fascinating concept, allowing us to generalize functional iteration to arbitrary real and complex iteration indices. We have $$ \begin{equation} \begin{split} S_f(0) & ...
0
votes
0answers
9 views

the bilinear mapping that maps the given three points onto the three given points in the respective order 1,i,-5 onto i,-2i,2

i need to solve this question : the bilinear mapping that maps the given three points onto the three given points in the respective order 1,i,-5 onto i,-2i,2 , the way i can think of is mobius ...
1
vote
3answers
31 views

imaginary algebraic inequality equation

This problem was actually given to me as a typo. I decided to work it despite it being a typo and it presented a couple of questions regarding applying imaginary results to an inequality equation. ...
0
votes
1answer
31 views

Finding modulus and argument of a complex number

I am having troubles with finding and argument of these two $$\frac{i}{1}$$ and $$\frac{2^{e^{i \theta}}}i $$ for the first one my approach was $$|z|=\frac{1}ie^0$$ $$e^{i\theta}=e^0$$ ...
3
votes
1answer
76 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
13 views

Proving that and LC of solutions is still a solution

I am currently using Lay's Lineair algebra and its functions, on page 316. On this page, I have the following problem. One page earlier is stated that a multiplication x' = Ax (where A is a matrix ...
1
vote
4answers
105 views

Derivative of a quadratic form

There is a Hermitian matrix $X$ and a complex vector $a$. I know that $a^HXa$ is a real scalar but derivative of $a^HXa$ with respect to $a$ is complex, $$\frac{\partial a^HXa}{\partial a}=Xa^*$$ Why ...
-1
votes
3answers
55 views

Describe the solutions of the equation in terms of roots of unity?

I want to find the solutions of the equation $$\left[z- \left( 4+\frac{1}{2}i\right)\right]^k = 1 $$ in terms of roots of unity. When I try to solve this, I get \begin{align*}z - 4 - \dfrac i2 ...
2
votes
2answers
165 views

Difference between i and -i

Consider the two imaginary numbers $i$ and $-i$. Is there any fundamental difference between the two of them? If I take the field $\mathbb{C}$ and apply the map $a + bi \mapsto a - bi$ does the image ...
5
votes
0answers
120 views

How can $ i $ be distinguished from $ - i $? [duplicate]

Mathematicians designate one solution to $x^2 = -1$ as $i$ and the other as $-i$. Would anybody notice if we switched their identities? Any polynomial $p(x)$ with a complex root will also have its ...
1
vote
1answer
66 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 ...
0
votes
1answer
34 views

Checking where the complex derivative of a function exists

I have the following function: $$f(x+iy) = x^2+iy^2$$ My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we ...
0
votes
1answer
101 views

Mandelbrot sets and radius of convergence

While watching this Numberphile video on Mandelbrot sets, it's more or less stated that the fractal will "blow up" if it's radius of convergence is greater than 2. What is the mathematical basis for ...
0
votes
1answer
44 views

How to build $\mathbb{C}$

I've defined $\mathbb{C}$ as $\mathbb{R} [X]/ (X^2+1)$, how do I show that $\mathbb{Q} [X]/ (X^2+1)$ is a subset of $\mathbb{C}$? And is $i \in \mathbb{Q} [X]/ (X^2+1)$? And can we see $\mathbb{Q} ...
3
votes
1answer
54 views

Drawing complex numbers on an argand diagram

I need some help drawing the following loci (which are rather hard to comprehend for me how will they look like) on an argand diagram: $$\arg \frac{i-z}{z+i}=\frac{\pi}{2} $$ (this one I suppose is ...
1
vote
1answer
44 views

Quadratic formula for $z^2 + (\alpha + \beta i)z + \gamma + i\delta = 0$ where $z\in\mathbb{C}$

The problem statement is to solve the quadratic equation $$ z^2 + (\alpha + i\beta)z + \gamma + i\delta = 0. ...