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

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3
votes
2answers
49 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
305 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
17 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
32 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
29 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
164 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
49 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
684 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
159 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
15 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
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
6 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
29 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
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 $$|z|=\frac{1}ie^0$$ $$e^{i\theta}=e^0$$ ...
3
votes
1answer
69 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
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 ...
1
vote
4answers
81 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
47 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
147 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
114 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
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 ...
0
votes
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
votes
1answer
89 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
41 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
47 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 ...
307
votes
18answers
55k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
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 ...
2
votes
3answers
132 views

In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?

Given dual numbers, what would be the value of $0^\varepsilon$ and $\varepsilon^\varepsilon$?
0
votes
0answers
28 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 $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ ...
6
votes
3answers
130 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
votes
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: ...
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 ...
1
vote
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
vote
1answer
26 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
votes
1answer
50 views

$\sin\left(1+\frac{1}{z-1}\right)$ expanded in powers of $z-1$

The whole problem: Obtain the following Laurent expansions. State the first four nonzero terms. State explicitly the $n$th term in the series, and state the largest possible annular domain in which ...
0
votes
1answer
72 views

When equality holds in an inequality

I am working on a class project, the passage I quoted in here is from a book Complex Numbers & Geometry by Hahn, p.64. For any four complex numbers $a$, $b$, $c$, $d$, the following identity ...
0
votes
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
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 ...
0
votes
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 ...
7
votes
1answer
79 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 ...
3
votes
1answer
28 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.
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
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 ...
4
votes
3answers
113 views

Geometry of the dual numbers

A dual number is a number of the form $a+b\varepsilon$, where $a,b \in \mathbb{R}$ and $\varepsilon$ is a nonreal number with the property $\varepsilon^2=0$. Dual numbers are in some ways similar to ...
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 ...
-2
votes
7answers
428 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
0
votes
1answer
29 views

Magnitude of a complex expression

Is there a way to derive an expression for the magnitude of $$ \frac{2 + (1-2ia\lambda \sin \theta)^{1/2}}{3 + 2ia\lambda\sin\theta} $$ I know how to do this if the square root weren't there. Any ...
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 $$x^3 + (1-i)x^2 + (1-i)x - i$$ ...
2
votes
3answers
68 views

Find the set of complex numbers $z$ which satisfy: $\left\lvert\frac{z-3}{z+3}\right\rvert=2$

Find the set of complex numbers $z$ which satisfy $$\left\lvert\frac{z-3}{z+3}\right\rvert=2\text.$$ I need help on that one. Thank you.