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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0
votes
1answer
8 views

Show that a line is tangent to a circle in the extended complex plane.

The straight line $l$ in the extended-complex plane pasess through $2+i,2+2i$.The circle $C$ centered at $-1-2i$ with radius $3$. First, I find the parametrization of the straight line which is $$z = ...
3
votes
1answer
107 views

Has anyone ever explored $(\sin{x})^x$ , $(\cos{x})^x$, etc?

I've come across a problem that involves something very close to: $$\int(\cos{x})^xdx$$ and I have no clue as to how to proceed with any kind of analysis for this type of equation. It occurred to me ...
40
votes
7answers
3k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

I know there must be something unmathematical in the following but I don't know where it is: \begin{align} \sqrt{-1} &= i \\ \\ \frac1{\sqrt{-1}} &= \frac1i \\ \\ \frac{\sqrt1}{\sqrt{-1}} ...
0
votes
0answers
35 views

Is the square root of -1 really “i” [duplicate]

I know that the imaginary unit i is a number with the following property: i^2 = -1 But I often see people turn that into this ...
1
vote
1answer
92 views

Proving that a complex number lies on the imaginary axis.

Given that there are two complex numbers - $z, w$ - such that $w\overline{w} = 1$ and $z = \frac{1+w}{1-w}$, how do I deduce that $z$ lies on the imaginary axis?
1
vote
1answer
22 views

Expression of reflection isometry in the complex plane

Using the fact that an anti-displacement in the plan has the form $$f(z) = a \overline{z} + b$$ I have done some computation to find the reflection about the line passing through two points $P$ and ...
2
votes
1answer
24 views

Doubt in raising a power to a complex number

What's the value of $$i^{i^{i^{...}}}$$? I tried to take log on both sides. But I could realise that I can't go with that.
1
vote
1answer
34 views

Find the number of elements of a complex subset

How many elements does the set $\{z\in \mathbb C:z^{60}=-1,z^k\not=-1\text{ for } 0<k<60\}$ have ? $z^{60}=-1=\cos(2k\pi+\pi)+i\sin(2k\pi+\pi)$. Then , ...
1
vote
1answer
35 views

An alternative method to find $\sum_{k=1}^{2n-1} | \beta ^k - 1|$

Let $\beta \in \mathbb{C}$ such that $\beta ^n = 1$ but $ \beta ^k \neq 1$, $\forall k=1,2,\cdots, n-1$. Find the value of $$ \sum_{k=1}^{2n-1} | \beta ^k - 1|.$$ I came across such a question ...
1
vote
1answer
18 views

Let $z,w\in \mathbb{C}$ where $|w|<1$ (modulus). What is the set of all $z\in \mathbb{C}$ that satisfies $|z-w|\leq|1-\bar{w}z|$.

Let $z,w\in \mathbb{C}$ where $|w|<1$ (modulus). What is the set of all $z\in \mathbb{C}$ that satisfies $|z-w|\leq|1-\bar{w}z|$. I've tried a few things with no luck. I wrote $z,w$ are complex ...
6
votes
2answers
95 views

Minimum of $f(z) = \left|z^2+z+1\right|+\left|z^2-z+1\right|$

For $z\in\mathbb{C}$, calculate the minimum value of $$ f(z) = \left|z^2+z+1\right|+\left|z^2-z+1\right| $$ My Attempt: Let $z= x+iy$. Then $$ \begin{align} z^2+z+1 &= ...
1
vote
1answer
33 views

Is it necessary for the Imaginary-axis to be perpendicular to the Real-axis?

The Real number line is in one dimension. If you want to map a complex number, you would have to add a second dimension to that number line- the Imaginary-axis. The Imaginary-axis is always ...
0
votes
2answers
24 views

calculating complex numbers - help needed [on hold]

Probably it is simple, but I am blind right now and I do not see how to solve this task: $e^{i \frac{2\pi}{3}}+e^{i\frac{4\pi}{3}}+1$
2
votes
2answers
58 views

a matrix of rank $r$ satisfies a polynomial of degree $r+1$.

Let $M$ be an $n\times n$ matrix with coefficients in $\mathbb C$. Suppose $M$ has rank $r$ with $r<n$. Prove there is a polynomial $P(x)$ with degree $r+1$ and coefficients in $\mathbb C$ such ...
0
votes
1answer
71 views

complex number passing from $|z|^{2}$ to $|z|$

I really couldn't understand last part when they pass from $|z|^{2}$ to $|z|$ so any more explanation please?
1
vote
1answer
122 views

Analytical Geometry problem with complex numbers - alternate solutions.

The question is to show that the equation of the lines making angles $45^\circ$ with the line: $$ \bar{a}z + a\bar{z} + b = 0; \;\;\;\;\; a,z \in \mathbb{C}, b \in \mathbb{R} $$ and passing through a ...
3
votes
1answer
402 views
+200

The image of a map $H^2 \setminus \Delta \to \mathbf{R}^4$ where $H$ is the upper half-plane and $\Delta$ is the diagonal

Let $H = \{z \in \mathbf{C}: \operatorname{Im} z > 0\}$ be the upper half-plane, and let $D = H^2 \setminus \Delta$, where $\Delta = \{(u,u) \in H^2\}$ is the diagonal. Define $\varphi: D \to ...
-2
votes
3answers
81 views

Find all complex numbers satisfying $x^4+x^2+1=0$ [on hold]

Find all solutions that fit: $$x^4+x^2+1=0$$ I did it couple of days ago, now I can't remember.
0
votes
3answers
450 views

Find the cube roots of $ -8 i $ and plot them on a plane.

I can’t figure out the angle of this equation. I set it up like this: $$ z^{3} = 0 - 8 i. $$ I find that the $ r $-value is $ 2 $, but when I try to find the angle, I’m stuck. I can’t divide by $ 0 ...
1
vote
1answer
18 views

Solve complex equation graphically

I have this problem that is split in 2, A and B, and Im struggling with B in particular, but I also dont know if I have done A correctly, which I suppose is necessary.. A) "Let $z$ be the complex ...
14
votes
1answer
5k views

What's more common? Re / Im or Fraktur-R / Fraktur-I for real / imaginary part?

Title says it all. What's more common? Is there one to prefere (maybe due to some norm)? This: $\operatorname{\mathfrak{R}} z, \operatorname{\mathfrak{I}} z$ or that: $\operatorname{Re}z, ...
1
vote
2answers
107 views

Show that $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ represents a circle

Suppose $|a|<1$ and $r\in (0,1)$. Show that the set of complex number $z$ satisfying $\left|\dfrac{z-a}{1-\bar a z}\right|=r$ is a circle in complex plane. Find the centre and radius of this ...
1
vote
2answers
22 views

Solve Complex Equation with several terms

I have a complex number $z = 3 + 3i$ And I want to find all solutions of $z^{10} + 2z^{5} + 2 = 0$ I'm kinda lost. I recognise the fact that I can substitute $u = z^{5}$ and rewrite the equation as ...
6
votes
3answers
181 views

Does there exist an analytic function $f$ such satisfying the following three conditions?

Does there exist an analytic function $f:\{z\in \mathbb C:|z|<1\}\to \{z\in \mathbb C:|z|<1\} $ such that, $f(0)=1/2$ , $f(1/2)=1/3$ , $f(1/3)=1/4$ ? I tried through the Schwarz-Pick lemma ...
0
votes
0answers
32 views

Fourier series for a Sinusoid in a conventional way?

So my TA in class introduced this amazing way of finding fourier series coefficients for a sin wave, by writing $ sin( \omega t ) = (e^{i\omega t}-e^{-i\omega t}) / 2i $ ----(1) Hence getting the ...
0
votes
1answer
394 views

Graphing Complex Number on Argand Diagram

Can someone please answer me, HOW does Im(z^2) = 4 get graphed like this? and not like a normal parabola? Like Re(z^2) = 4 But of course on the Imaginary axis. It has been eating my mind up - ...
2
votes
4answers
110 views

Is $i^i$ mathematically valid? [duplicate]

WARNING: SLIGHT NSFW http://www.smbc-comics.com/index.php?db=comics&id=2934#comic Uhh...guys, mathematically speaking, how accurate is this comic. From what I remember in High School $$a^b= ...
0
votes
1answer
37 views

Exponential Complex Number

I need assistance in solving the following: http://i.stack.imgur.com/EcGLD.jpg I am not very sure on how to remove the exponential to convert it into complex numbers and get the arguments in the ...
0
votes
0answers
28 views

Is there a name for complex numbers over affinely extended reals?

Is there a name for the set of complex numbers over affinely extended real line, that is $\mathbb{C}\cup \{-\infty\}\cup\{+\infty\}$? I think this set is the most commonly used in analysis ...
0
votes
3answers
37 views

$|z|=1$ should represent semi-circle or circle?

Suppose we have complex number $z=x+iy$ and we are given locus $|z|=1$ which should be $\sqrt{x^2+y^2} =1$ this should be a semi-circle above x axis , it's when we square our equation we get a circle ...
2
votes
1answer
39 views

Find the min and max distance from origin of the curve $\vert z+\frac{1}{z}\vert=a$

$z$ is a complex number, by the way. I've tried a lot of things and always end up with a huge algebraic mess and I've wondered if anyone of you has any idea on how to approach this problem. One of ...
1
vote
2answers
38 views

multiple sets of complex roots of a number?

I am not sure if this question was asked before but I couldn't find the right keywords to choose for searching. So today I discovered a weird problem: If we take this equation: $$x^2=1=e^{(0i)}$$ ...
0
votes
2answers
48 views

find the coefficient

If $n$ is an odd natural number, and $\sin(n\theta) = \Sigma_{r=0}^{n} b_r \sin^r\theta$, then find $b_r$ in terms of $n$. I have tried this using trigonometric expansion but unable to find solution ...
1
vote
1answer
35 views

Find the mistake in calculation [duplicate]

$(-1)^3 = (-1)^{6/2} = ((-1)^6)^{1/2} = 1^{1/2} = 1$ So it comes $(-1)^3 = 1$ can anybody explain where exactly the mistake in calculation?
0
votes
2answers
25 views

the locus of $z$ in the complex plane

Describe the locus of $z$ in the complex plane if $z$ satisfies: $$ arg(z)=arg(z+3+i)\quad (mod\ 2\pi) $$ Indeed Let $O$ be the origin and $B=-3-i$. \begin{align*} arg(z)&=arg(z+3+i)\quad ...
0
votes
3answers
60 views

Finding the modulus and argument of a complex number

I would need help with this question: $$Z = \frac{(1+j2)^2(4-j3)^3 }{ (3+j4)^4 (2-j3)}$$ My starting point for this question is to expand the complex numbers first then continue doing but after ...
0
votes
0answers
20 views

Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, then $u_n \rightarrow 0$ uniformly on compact subsets of $D$.

Let $D \subseteq \Bbb C$ be a connected open subset and let {$u_n$} be a sequence of harmonic functions $u_n: D \rightarrow (0,\infty)$. Show that if $u_n (z_0) \rightarrow 0$ for some $z_0 \in D$, ...
7
votes
1answer
175 views

Complex Numbers $\stackrel{?}{=} \mathbb{R}^ 2$

Suppose we have a vector field over real numbers $\mathbb R^2$. In additon to vector field proporties define inner product $(x,y) = x_1\cdot y_1 + x_2\cdot y_2$, where $x_1,x_2,y_1,y_2$ are real ...
2
votes
2answers
46 views

What is the interval for possible values of the argument of a complex number?

It looks like there are different intervals in which the argument of a complex number can be. Some say it goes from $-\pi$ to $+\pi$ others say it goes from $0$ to $2\pi$. For the most part, both ...
0
votes
3answers
41 views

Reducing to upper triangular form

I've just had some difficulty with this transforming this matrix into upper triangular form: $$ \pmatrix{ i& 2i& -1\\1 & 1& i\\ 2-i& 1& i } $$ I've tried almost everything. ...
1
vote
2answers
37 views

Express the complex number in rectangular form $a + ib$

$12e^{2-\pi*i/3}$ express this in rectangular form $a + i\cdot b$ Not sure how to solve when fractions are involved Example $2.6\cdot e^{3+i} = 2.6\cdot e^3\cdot e^i$ ?
0
votes
0answers
17 views

If $f:B_N\rightarrow \mathbb{D}$ and $z_n\in B_N$ with $\{f(z_n)\}$ thin, is $\{f(\phi(z_n))\}$ thin for any autmorphism $\phi$ of $B_N$?

Let $B_N$ denote the open unit ball in $\mathbb{C}_N$. A sequence $\{z_n\}$ of distinct points in $\mathbb{D}$ is called thin if $\lim_{k\rightarrow \infty}\displaystyle\prod_{j: j\not =k}^\infty ...
310
votes
35answers
36k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
-1
votes
2answers
63 views

2+2=square root of 16. What's the appropriate answer? [closed]

4? Positive and negative 4? I just got into an argument with a buddy about this. He argues if it's not an i, it's not included as a imaginary number, but only the real positive number.
3
votes
3answers
275 views

Finding complex number defined by 3 equations

Let $z$ be a complex number satisfying $$\DeclareMathOperator{\Re}{Re}\Re[z^4]=1/2$$ $$z\bar{z}+2|z|-3=0$$ $$\arg z \leq \frac{\pi}{4}.$$
11
votes
8answers
781 views

If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle

Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.) How do I prove this? I tried to use the dot product of 2 adjacent ...
0
votes
1answer
39 views

Compute the integral $\int_\mathbb{R^{n}}\lvert x\rvert ^{2m}\exp(- \lvert x\rvert^2/2) \mathrm{dx} $ [closed]

Please help me to compute the integral $\int_\mathbb{R^{n}}\lvert x\rvert^{2m} \exp(- \lvert x\rvert^2/2) \mathrm{dx} $, where $ m = 1,2,...$ using some complex analysis results. Thanks in advance
13
votes
5answers
1k views

Can I keep adding more dimensions to complex numbers?

I know about the concept of the complex plane, but is it possible to move to the third dimension? What about arbitrary many dimensions? Edit: could you please give me some examples of 3D numbers?
38
votes
4answers
8k views

Prove that $i^i$ is a real number

According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
1
vote
2answers
56 views

Powers of complex numbers property

I would like to prove the following statement. Let $\lambda_1,\dots,\lambda_s \in \mathbb{C}$ be such that $|\lambda_1| = \dots = |\lambda_s|=1$. Then $\forall \varepsilon \gt 0$ there exist ...