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

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1
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3answers
77 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
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2answers
27 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 ...
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1answer
12 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 ...
1
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2answers
36 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: $$ x_{1,2} = \frac{2a \pm \sqrt {4a^2-4b^2}}{2} = {a \pm \sqrt {a^2-b^2}} = a\pm \sqrt{a-b}\sqrt{a+b}$$ ...
0
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1answer
37 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 $x$?
4
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2answers
219 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, ...
2
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4answers
64 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 ...
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1answer
48 views

Is $i$ defined? [on hold]

The imaginary number $i$ is defined to be $\sqrt{-1}$, or the number satisfying $x^2+1=0$. But note that both $i$ and $-i$ solve that equation. How is it known whether $i$ is actually positive or ...
1
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1answer
38 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
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1answer
52 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 ...
2
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1answer
45 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 $M \in \mathbb{N}$ ...
1
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1answer
39 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 ...
0
votes
1answer
37 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 ...
37
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22answers
4k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
8
votes
2answers
104 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 ...
3
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2answers
62 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 ...
0
votes
2answers
19 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
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0answers
38 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
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1answer
30 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 ...
1
vote
1answer
52 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 $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ ...
0
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1answer
17 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) & ...
1
vote
2answers
163 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
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0answers
13 views

Harmonic function condition for $ v=f(x,y)$ [on hold]

I know that to see if $u=F(x,y$) is a harmonic function $Uxx+Uyy=0$ but if instead of U function I have the V function $v=F(x,y)$ is it still $Vxx+Vyy=0$ or I should check something else , ...
0
votes
0answers
7 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 ...
0
votes
1answer
29 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 $$|z|=\frac{1}ie^0$$ $$e^{i\theta}=e^0$$ ...
0
votes
1answer
10 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 ...
5
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0answers
117 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
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1answer
38 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
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4answers
92 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 ...
0
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1answer
29 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
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1answer
42 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
48 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 ...
0
votes
1answer
38 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. ...
0
votes
2answers
25 views

Looking for proof of theorem on complex measurable functions

In University I have been given the following result: If $f:X\to\mathbb{C}$ is a measurable function in $L^1(X,\mathcal{E},\mu)$ with $\mu$ being finite, and there exists a closed set ...
0
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0answers
32 views

Is split-complex $j=i+2\epsilon$?

In matrix representation imaginary unit $$i=\begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix}$$ dual numbers unit $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ ...
7
votes
3answers
134 views

How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form ...
0
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0answers
28 views

Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

I have this question: Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection? Here's what I tried: ...
1
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1answer
28 views

Solution for a complexed equation

Find $z$ for the equation $e^z + e^{-z} = 0$. So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$ I understand all expect the $2\pi ik$. Can you ...
1
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1answer
27 views

find roots in the complexes

Find the roots of: $$ z^2 -3z +4iz = 1-5i $$ Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $ Solving by using the quadratic formula: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
0
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0answers
19 views

Does $t^2y''-ty'+y=2t$ with $y(t)=tz(t)$ only if $z'$ as $t^2u'+tu=2$?

I have this question: We have $y$ and $z$ functions with reals as $y(t) = t\times{z}(t)$ for $t \in I = ]0,+\infty[$. Then $y$ satisfies $t^2\times{y}''-t\times{y}'+y=2t$ on $I$ if and only ...
0
votes
1answer
49 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
0
votes
1answer
30 views

Are $\Re(z)$ and $\Im(z)$ solutions of $z' = az$?

I'm having trouble with a question (I have to answer "true" or "false" and explain it): We have $a:I \to C,$ a continuous function on $I$, interval of $R$. If the function $z:I \to C$ is a ...
3
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1answer
29 views

Is i holomorphic over the whole complex plane?

That is, is i entire? I know that it's derivative with respect to z bar is 0, so I would think that the answer is yes, although I'm not sure.
7
votes
1answer
80 views

Does $\exp(2ir\pi)$ equal $1$? What's wrong?

Since $e^{ix}=\cos x+i\sin x$, thus $e^{2\pi i}=\cos2\pi+i\sin2\pi=1$ Now I take arbitrary real number $r$ then $e^{i2\pi r}=(e^{i2\pi})^r=1^r=1?$ But this cannot be true since $\cos2\pi ...
1
vote
0answers
23 views

Finding extreme complex numbers satisfying a condition

Let $a$ be a positive real number and let $$M_a = \left\{z \in \mathbb{C^*}: \left|z + \frac{1}{z}\right| = a\right\}$$ Find the minimum and maximum value of $|z|$ when $z\in M_a$. ($\mathbb{C^*}$ ...
0
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0answers
32 views

Chain rule (derivative) for for complex data

I found some difficulties in extending the chain rule for complex data. Any suggestion will be appreciated, thanks. In the complex domain, for example, we have a function ...
0
votes
2answers
35 views

to find radius of convergence of power series.

I have a power series given as: $f(z) =1 + z+ \frac{z^2}{2^2} +\frac{z^3}{3!} + \frac{z^4}{2^4} \frac{z^2}{2^2}+ \frac{z^5}{5!}+ \ldots$ I have to find radius of convergence of above series. My ...
1
vote
1answer
53 views

how to solve complex integration problem

While working on complex integration problem I got stuck at the following problem: $\int \frac{|dz|}{|z-2|^2}$ where $|z| = 1$ is the domain. The only idea that I am getting is that we can use the ...
0
votes
1answer
32 views

Complex roots of Complex polynomal

Apologies if this is a repeated thread I just couldn't quite find anything that helped. how do I go about finding the complex roots of a complex polynomial? such as $$x^3 + (1-i)x^2 + (1-i)x - i$$ ...
-1
votes
3answers
51 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 ...