Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
4
votes
2answers
180 views
An expression involving p-th root of unity
Let $\zeta_p$ be a $p$-th root of unity, where $p$ is an odd prime number.
I just came across the following expression:
$$\frac{(\zeta_p^2-\zeta_p+1)^3}{\zeta_p^2(\zeta_p-1)^2}.$$ Can we simplify ...
4
votes
2answers
90 views
Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $
I'm trying to find
$$\sum _{k=1}^{n} k\cos(k\theta)\qquad\text{and}\qquad\sum _{k=1}^{n} k\sin(k\theta)$$
I tried working with complex numbers, defining $z=\cos(\theta)+ i \sin(\theta)$ and using ...
4
votes
1answer
152 views
Rudin's Proof of Cauchy-Schwarz (Theorem 1.35)
Any motivation for the sum that Rudin considers in his proof of the Cauchy-Schwarz Inequality?
1.35 Theorem If $a_1,...,a_n$ and $b_1, ..., b_n$ are complex numbers, then
$|\sum_{j=1}^n ...
4
votes
2answers
72 views
Complex numbers: With conjugate
I've just started calculating complex numbers (last time I calculated with complex numbers was in high school) and I've already got stuck at this exercise:
$$3z-i\bar z = 7-5i$$
where $\bar z$ ...
4
votes
3answers
143 views
Complex series: $\sum_{n=0}^\infty\left( z^{n-2}/5^{n+1}\right)$ for $0 < |z| < 5$
How would one compute
$$
\sum_{n=0}^\infty\frac{z^{n-2}}{5^{n+1}}
$$
where $0\lt|z|\lt5$?
I have literally no idea where to start, all I know is that the answer will not have summations. Any help ...
4
votes
2answers
426 views
If z is one of the fifth roots of unity, not 1…
If z is one of the fifth roots of unity, not 1, show that:
$1+z+z^2+z^3+z^4=0$
Which wasn't too bad, but the next part is killing me: show that:
$z-z^2+z^3-z^4=2i(sin(2\pi/5)-sin(\pi/5))$
Can ...
4
votes
3answers
133 views
High school math problem: Trouble with complex representation
What is $\sum\limits_{n>0,\text{ odd}} r^n \sin(nx)$ in terms of $z=re^{ix}$? I tried to write $\sin(nx)={e^{inx}-e^{-inx}\over 2i}$ but then I have a sign problem because the $n$ on the associated ...
4
votes
1answer
149 views
If $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then $|z| \leq (4 \sin(1/4))^{-1} |\sin z|$
I came across the following assertion and am having trouble justifying it:
If $z$ is a nonzero complex number with $|z| \leq \pi/2$ and $|\sin z| \leq 1/4$, then
$$
\left| \frac{z}{\sin z} \right| ...
4
votes
1answer
121 views
Well-ordering of positive Gaussian integers under lexicographical ordering?
I am reading a paper by Richard Weimer called "Can the complex numbers be ordered?" and he makes the following claim.
Let $G^+=\{a+bi : a,b$ are positive integers $ \}$ and let $<$ denote ...
4
votes
1answer
19 views
Complex number equivalency
I'm a bit confused over the solution to a complex ode: $i\alpha y = \beta y''$
The solution to the characteristic polynomial is $r = \pm \sqrt{i\alpha/\beta}$. Somehow my book is getting the ...
4
votes
1answer
301 views
Why are imaginary numbers called imaginary numbers
Why do we call imaginary numbers "imaginary numbers"? As far as I can tell, there's nothing really imaginary about them. They exist. They're used all the time. What makes them so "imaginary"?
4
votes
2answers
236 views
Do “complex percentages” exist?
Well, the origin of this question is a little bit strange. I dreamed - with a book called "Percentages and complex numbers. When I woke up, I thought: "Is this real?" So I started thinking:
...
4
votes
3answers
89 views
How do you construct the quaternion and the multiplication rules, like Hamilton did?
So, I understand complex number multiplication, and how it represents $2D$ rotations.
What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
4
votes
2answers
180 views
Difference between imaginary and complex numbers
Recently I was talking to my teacher about complex and imaginary numbers and he told me basically that $i$ is a complex number; its real part is just 0. However, this has made me wonder; if you can ...
4
votes
3answers
215 views
Applications of complex variables beyond undergrad syllabus
So complex numbers solve all polynomials, appear as eigenvalues, appear in intermediate calculations in solving cubics, relate trig to hyperbolic functions, can be used to contour integrate real ...
4
votes
1answer
272 views
Using the complex logarithm to find the sum of angles in a triangle.
Suppose you have a triangle with vertices $a$, $b$, and $c$. I asked earlier how you can define the angles in a triangle based on the $\log$ function. I received the answer that, for instance, the ...
4
votes
1answer
86 views
Is there a $z$ for which $z$, $1+i$, $(1+i)z$ and $e^z$ are collinear?
Is there a $z$ for which $z$, $1+i$, $(1+i)z$ and $e^z$ are collinear? There is a close call around $z = .18 + 1.09i$ but I'd like to see a mathematical solution.
4
votes
1answer
129 views
Modulo complex number
I was wondering what would happen if we tried to do a modulo operand with complex numbers? For instance, what would be the answer (if any) to the next statement?
$ x \mod (a + bi) $
can it be ...
4
votes
1answer
106 views
When $|f|$ and $\arg(f)$ are analytic?
Let $f:ℂ→ℂ$ be an analytic function.
Define $|f|$ and $\arg(f)$ be the modulus and the argument of $f$. Generally, $|f|$ and $\arg(f)$ are not analytic.
My question is about the cases where this ...
4
votes
2answers
126 views
Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?
I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
4
votes
2answers
100 views
Quick Julia/Mandelbrot Testing
I have successfully implemented a realtime Julia/Mandelbrot set generator on the GPU. Primarily out of curiosity, what I'm looking for now is a faster test algorithm.
Ideally, I want a boolean ...
4
votes
2answers
256 views
Is a 3D Mandelbrot-esque fractal analogue possible?
I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties).
Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
4
votes
1answer
43 views
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$
Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by ...
4
votes
1answer
116 views
How to calculate $\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$? (second question)
I asked yesterday for a hint on how to calculate $$1+\sum_{k=1}^{k=n}\frac{\sin(kx)}{\sin^{k}(x)}$$
I worked on this problem for another couple of hours and now I am stuck again, I would greatly ...
4
votes
1answer
108 views
About problem in complex integrals
I solved this problem in complex integrals.
Is my answer a correct ?
Here $z$ is a complex value:
$$
C:|z-1|=1 \ \ \ \ \ \mbox{integral path}
$$
$$
\int_C\ \frac{2z^2-5z+1}{z-1}\ dz
$$
My answer
...
4
votes
1answer
185 views
What is the Jacobian?
What is the Jacobian of the function $f(u+iv)={u+iv-a\over u+iv-b}$?
I think the Jacobian should be something of the form $\left(\begin{matrix}
{\partial f_1\over\partial u} & {\partial ...
4
votes
1answer
289 views
Why is the argument of $i$ equal to $\pi/2$?
So it's obvious geometrically that the argument of $z=i$ is $\pi/2$.
However the method of getting the argument is $\arctan(y/x)$. And when in the case of $z=i$, $y/x = 1/0$ which is undefined...
So ...
4
votes
2answers
130 views
Problem finding zeros of complex polynomial
I'm trying to solve this problem
$$ z^2 + (\sqrt{3} + i)|z| \bar{z}^2 = 0 $$
So, I know $ |z^2| = |z|^2 = a^2 + b ^2 $ and $ \operatorname{Arg}(z^2) = 2 \operatorname{Arg} (z) - 2k \pi = 2 \arctan ...
4
votes
2answers
411 views
Using complex exponentials as solution of ODE
I'm having trouble wrapping my head around the following issue. My book solves a problem without using complex exponential solution like $C_1 e^{it}$ and using either $A \cos(t) + B \sin(t)$ or $A ...
4
votes
3answers
75 views
Different interpretations of imaginary number
I went through a linear algebra course and I'm a bit confused..
I think I understand the geometric interpretation of imaginary numbers - multiplying by $i$ results in rotation by $90$ degrees in so ...
4
votes
3answers
78 views
When does the next complex split occur?
So I was thinking about complex numbers and how they came about and someting interesting occured to me:
the formation of complex numbers occurs because there exists a function (namely $f(x)=x^2$) ...
4
votes
2answers
102 views
Proof $\int\Re(f(x))\,\mathrm{d}x=\Re(\int f(x)\,\mathrm{d}x)$
I have a function $f: \mathbb{R}\to\mathbb{C}$. How can I proof/argue that $$\int\Re(f(x))\,\mathrm{d}x=\Re\left(\int f(x)\,\mathrm{d}x\right)$$ (and the same for the imaginary part)? I'm afraid I ...
4
votes
1answer
100 views
Existence problem for a polynomial with complex coefficients
Let $n$ be a nonnegative integer and $a_{0}, a_{1}, ..., a_{n}$ real numbers. For any real number $t$ let $f(t)= \sum_{k=0}^{n}a_{k}\cos(kt)$.
Could you help me with the following two questions ?
a) ...
4
votes
0answers
80 views
Natural extension of the divisor function $\sigma$ to the complex integers $\mathbb{Z}[i]$
The divisor function $\sigma$ of a positive integer $n$ is defined as the sum of the positive divisors of $n$. It turns out that this definition is equivalent to
...
4
votes
0answers
55 views
Simplest examples of real world situations that can be elegantly represented with complex numbers
Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
4
votes
1answer
67 views
What are the uses of split-complex numbers?
The set of Complex numbers can be used in lots of domains like geometry, vectorial calculations, solving equation with no real solution etc. But what are the uses of split-complex number that can't be ...
4
votes
0answers
243 views
What are applications of Lagrange's identity?
I recently proved for homework the following identity on $\mathbb{C}$: if $a_1, \ldots , a_n, b_1, \ldots, b_n\in\mathbb{C}$, then
$$
\left|\sum_{i=1}^na_ib_i\right|^2 = ...
4
votes
1answer
124 views
Prove that there exists analytic $f$ such that $f(z) = 1/\bar{z}$ on the boundary
I'm doing some self-study in complex analysis, and came to the following question:
Let $D(a,1) \subset \mathbb{C}$ be the disk of radius $1$ with center at $a \in \mathbb{C}$, and let $\partial ...
4
votes
1answer
794 views
Proving the Schwarz Inequality for Complex Numbers using Induction
I want to prove the following version of the Schwarz Inequality for complex numbers $a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $b_1, b_2, \ldots, b_n \in \mathbb{C}$:
$$|\sum_{j=1}^n a_j ...
4
votes
1answer
213 views
Something about $i^i$ [duplicate]
Possible Duplicate:
What is the value of 1^i?
Note that I am absolutely not a mathematician, so this may be silly, but I saw this on Wikipedia's page about $i$:
One definition of $i^i$ ...
4
votes
2answers
369 views
What is the formula for the first Riemann zeta zero?
I found this approximation of which an earlier version I posted in the chat room:
$$7 \pi -\text{Log}\left[\frac{7}{2} e^{-7 \pi /2}+\frac{5}{2} e^{-5 \pi /2}+\frac{3}{2} e^{-3 \pi /2}+e^{5 \pi /2}+2 ...
3
votes
7answers
292 views
Show that $z^{-1} = \frac{\bar z}{|z|^2}$
I'm stuck on this question, I have a feeling the answer is very straightforward but I just can't figure it out.
Question: Considering $z= x + iy$, show that: $$z^{-1} = \frac{\bar z}{|z|^2}$$
So ...
3
votes
4answers
217 views
How $m^4+4 = 0 \Rightarrow m = 1 \pm i,-1\pm i$?
This is given in my module as a part of a problem's solution:
$$m^4 + 4 = 0 $$ $$\Rightarrow m = 1 \pm i,-1\pm i$$
I am not getting how this conversion is taking place,could somebody explain?
3
votes
3answers
463 views
Euler formula and $\sin^3$
Using the formula:
$$e^{i\omega t} = \cos {\omega t} + i\sin{\omega t}$$
I would like to prove that:
$$\sin^3\;x = -\frac{\sin{3x} - 3\sin{x}}{4} $$
However I haven't found any approach ...
3
votes
6answers
243 views
Proof that $e^{i\pi} = -1$
When I first found out that $e^{i\pi} = -1$, I was blown away. Does anyone here know one of (many I'm sure) proofs of this phenomenal equation? I can perform all of the algebra to get the $-1$. But, ...
3
votes
2answers
305 views
No extension to complex numbers?
Complex numbers are 2D. It is a commonly sited result that there is no 3D or 4D analogue of the complex numbers.
I just want to be clear on exactly what this result says:
It is impossible to ...
3
votes
2answers
214 views
Is it standard to say $-i \log(-1)$ is $\pi$?
I typed $\pi$ into Wolfram Alpha and in the short list of definitions there appeared
$$ \pi = -i \log(-1)$$
which really bothered me. Multiplying on both sides by $2i$:
$$ 2\pi i = 2 \log(-1) = ...
3
votes
3answers
84 views
What does the square root of minus $i$ equal?
Can you enter the rabbit hole recursively?
If the $ \sqrt{-1} = i $ then, what does $ \sqrt{-i} $ equal?
3
votes
3answers
205 views
Proof of $\sin^2 x+\cos^2 x=1$ using Euler's Formula
How would you prove $\sin^2x + \cos^2x = 1$ using Euler's formula?
$$e^{ix} = \cos(x) + i\sin(x)$$
This is what I have so far:
$$\sin(x) = \frac{1}{2i}(e^{ix}-e^{-ix})$$
$$\cos(x) = \frac{1}{2} ...
3
votes
4answers
352 views
Calculate a sum involving nth root of unity
Calculate
$$1+2\epsilon+3\epsilon^{2}+\cdots+n\epsilon^{n-1}$$
Where $\epsilon$ is nth root of unity.
There is a hint that says: multiply by $(1-\epsilon)$
Doing this multiplication I get:
...
