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

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1answer
28 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
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1answer
68 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 $$ ...
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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 ...
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4answers
77 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 ...
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3answers
45 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
136 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 ...
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0answers
113 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 ...
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1answer
33 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
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1answer
28 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
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 ...
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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
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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
24 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
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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 $\varepsilon^\varepsilon$?
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0answers
26 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 $$\epsilon=\begin{pmatrix}0 & 1 \\ 0 & 0 \end{pmatrix}$$ ...
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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 ...
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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: ...
0
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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 ...
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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 ...
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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: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
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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 ...
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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 ...
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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
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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 ...
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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 ...
7
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1answer
77 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
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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.
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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
112 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 ...
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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 ...
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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
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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
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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
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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.
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0answers
28 views

For which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$? [closed]

$ x $ and $y$ are real numbers and $i$ : is unit imaginary part . 1-for which values of $n$ does $x^n+y^n=i$ has a zeros in $\mathbb{R}$ ? 2-what are the possible geometrics forms of $x^n+y^n=i$ ...
3
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2answers
50 views

Proving that for all complex $z$, $\lim_{x\to0}\frac{1-\cos^{z}x}{x^2}=\frac{z}{2}.$

What do I need to study beforehand in order to prove it (not necessarily in only one way)? I found this sperimentally, at the moment we're beginning derivatives at school. By induction, I succeeded in ...
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2answers
56 views

How to change the limits of a summation when the index $k$ is replaced by $-k$?

Is what I am doing below correct, please assist. $$\sum_{k=-\infty}^{-1}\frac{e^{kt}}{1-kt} = \sum_{k=1}^{\infty}\frac{e^{-{kt}}}{1-kt}$$ Is this the rule on how to "invert" the limits, and does ...
2
votes
3answers
65 views

If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$

Question : If $|z-3i|+|z-4|=5$ then find the minimum value of $|z|$ What I did : $$|z-3i| \leq |z|+3 \tag i$$ Also $$|z-4| \leq |z| +4 \tag{ii}$$ Now adding (i) and (ii) we get $$ ...
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2answers
50 views

Find all solutions for $z^3 = \overline{z}$

I know that $z = a + ib$ and that $\overline{z} = a - ib$, but when I try and calculate the solutions I get an unsolvable equation. Would appreciate any help.
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1answer
29 views

Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ …

Problem : Let $|z|=1, $ prove that $|z^2-3z+1|\leq 5$ My approach : Let $z = x +iy$ $ \Rightarrow (x^2+y^2)=1$ $\Rightarrow |z| =1 $ represent a circle with centre at (0,0) and radius 1 ...
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0answers
17 views

Diffferentiability of complex functions

I need you help me please. I don't know how solve this Find $f_{z}$ y $f_{\bar{z}}$ where $f(z)=\left |{z}\right |^{2} +\displaystyle\frac{z}{\bar{z}}$ moreover what points is differentiable f ? ...
3
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4answers
135 views

Minimum value of $|z+1|+|z-1|+|z-i|$

How to find the minimum value of $|z+1|+|z-1|+|z-i|$. I have tried geometrically etc but failed.
0
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1answer
31 views

inequality in absolute value with exponential

Can you help me with this problem please ?? If $r>0$ . Show that $\left |{\displaystyle\int_{\gamma}e^{iz^{2}}dz}\right |\leq{\displaystyle\frac{\pi(1-e^{-r^{2}})}{4r}}$ where ...
0
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1answer
41 views

how to prove that a function is not complex differentiable

I was working on a problem on the complex differentiability of the following function: $f(z)= z \operatorname{Re}(z)$. How to find the points where the given function is not differentiable. My ...
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1answer
26 views

How to find $\lim\limits_{z \to z_0} \frac{{\overline z}^2-{\overline z_0}^2}{z-z_0}$

Fairly simple question, I want to find this limit $$\lim_{z \to z_0} \frac{{\overline z}^2-{\overline {z_0}}^2}{z-z_0}$$ The original question was to find the region at which the function ...
1
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4answers
38 views

Can we add fractional powers of negative numbers?

This question might be silly and very basic. But my friend and me happened to argue on this for long. My argument was, if $-2 \sqrt3=\sqrt{12}$ which came from $\sqrt{(-2)(-2)} \sqrt{3} $ . If this is ...
0
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0answers
55 views

Finding complex eigenvectors of $n \times n$ matrix, $n\geq 3$

An example: $$ \begin{pmatrix} 1 & 2 & 0 \\ 2 & -3 & 4 \\ 4 & -8 & 7 \\ \end{pmatrix} $$ Has eigenvalues $3$, $1+2i$, $1-2i$. How ...