Tagged Questions
6
votes
4answers
146 views
If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?
Is the following true if $\theta\in\mathbb{Q}$?
$$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$
Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
1
vote
2answers
53 views
Do any issues arise if we try to raise an element of $\mathbb{R}^+$ to an element of $\mathbb{C}$?
If $a$ and $b$ are non-zero natural numbers, the definition of $a^b$ is clear. Now it seems to me that there are (at least) two distinct ways of generalizing to larger number systems.
Firstly, given ...
0
votes
1answer
54 views
Why $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega t})$
Let $i := \sqrt{-1}$, $f$ be the frequency ($\frac1p$), and $\omega := 2 \pi f$.
From page 3 here, why does $\frac{1}{2i}(e^{i\omega t} - e^{-i\omega t}) = \frac{i}{2} (e^{-i \omega t} - e^{i\omega ...
0
votes
1answer
32 views
Put the following in rectangular form.
$$(\sqrt{3}+i)^7$$
My question: $r = 2$. For $\theta$, do I use $\dfrac{\pi}{6}$ or $\dfrac{\pi}{6} + 2n\pi$? The book uses the former but I thought the latter is more appropriate.
Thank you.
0
votes
0answers
23 views
check validity of following manipulation
in my algebra book,there is written following well known identity
$e^{2*\pi*i}=1$
generally we can use also this identity $e^{k*\pi*i}=(-1)^k$
and if instead of $k$,we put $2$ we get ...
24
votes
14answers
2k views
How do I understand $e^i$ which is so common?
Raising something to an imaginary number is weird, I have a hard time wrapping my head around that.
And e seems even more common and comes up in many situations, such as:
the non-geometric ...
4
votes
3answers
116 views
Where is the mistake in this proof?
I can't figure out, where is the mistake:
$$z=re^{i\phi}=re^{\large \frac{2\pi i\phi}{2\pi}}=r(e^{2\pi i})^{\large\frac{\phi}{2\pi}}=r1^{\large\frac{\phi}{2\pi}}=r1=r$$
And we found that the complex ...
5
votes
7answers
447 views
Is $0^0=1$ postulate independent of all other axioms of complex numbers?
This question is inspired by the other question which asked for a proof that $i^i$ is a real number.
Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
22
votes
1answer
280 views
Iterated exponent of $i$
WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
0
votes
3answers
296 views
How complex exponential converges and “sum of exponents” rule holds
How is it the complex exponential converges for any value of $z$ in the complex plane? $$e^{z} = 1 + \frac{z}{1!} + \frac{z^2}{2!} \cdots\cdots$$
How is it the "sum of exponents" rule holds for ...
4
votes
2answers
132 views
Is the multiplicative complex plane a Lie group?
I know that the complex plane is a Lie group with +, but is it also a Lie group with the usual complex multiplication?
This would give us a nice geometrical interpretation of the famous Euler ...
2
votes
2answers
166 views
How to find complex numbers $z,\lambda,\mu$ such that $(z^\lambda)^\mu\neq z^{\lambda\mu}$
Let $z$, $\lambda$, $\mu$ be complex numbers. Find a case where $(z^\lambda)^\mu$ is not equal to $z^{\lambda\mu}$.
In our book, $a^b = \exp( b \cdot \operatorname{Log}(a) )$.
...
1
vote
2answers
278 views
Fixed points of $e^z$
How would one find the fixed points of $e^z$, where $z$ is complex (if there are any)? I feel this problem probably has a really obvious answer, and for some reason, I'm just not getting it. Thanks.
-2
votes
1answer
162 views
Smallest positive integer for equation
I am having trouble identify the smallest positive integer $n$ such that $(\frac{1+i}{1-i})^n = 1$
Can someone please throw on approach?
(Also, please correct the equation in the form of Tex/Latex ...
41
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
10
votes
4answers
435 views
Which step in this process allows me to erroneously conclude that $i = 1$
I was playing around with imaginary numbers and exponents and came up with this:
$$ i = \sqrt{-1} $$
$$ \sqrt{-1} = (-1)^{1/2} $$
$$ (-1)^{1/2} = (-1)^{2/4} $$
$$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...
1
vote
1answer
86 views
compute an exponential complex number
I have a pretty basic question about complex numbers.
If $z=x+yi$, a complex number, i want to compute the real and the imaginary part of the number $w=e^{e^z}$.
Thanks in advance for any help.
1
vote
3answers
246 views
A contradiction involving exponents
Where is the error in the following statement:
$i^2=(i^2)^{\frac{4}{4}}=(i^4)^{\frac{2}{4}}=(1)^{\frac{1}{2}}=1$?
I feel the error is in the first equality, because $(i^2)^{\frac{4}{4}}$ is in fact ...
8
votes
3answers
310 views
Proof for law of complex exponents using only differential equation
I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...
5
votes
6answers
330 views
Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?
In my ongoing strugle to understand $e^{\pi i}$ I managed to narrow down my conceptual difficulty. I'm having intuitive trouble understanding why $(1 + iX/n)^{n}$ is conceptually the same as a ...
2
votes
3answers
617 views
Complex Exponents
What does it mean to raise a number to a complex exponent, and why? A lot of the explanations that I've seen involve e, why is this?
I'm looking for an intuitive answer describing to how ...
24
votes
3answers
1k views
