I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave.
It's funny, I've seen almost this exact scenario happen once.
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NMcCoy wrote:Regarding the sine wave thing: Are you familiar with Taylor series? The Taylor expansion of sin(x) is the odd terms of the Taylor expansion of e^x with alternating signs reversed, and the Taylor expansion of cos(x) is the even terms with alternating signs. Thus e^(x*i) = sin(x) + cos(x); in the case of pi, -1.
Drostie wrote:the guy used j instead of i. He had the nasty habit of writing {%omega t - k x } instead of {k x - %omega t}.
Drostie wrote:I remember one of my E&M classes at Cornell was taught out of an electrical engineering book by Staelin et al. ... the back cover was typo-ridden and the guy used j instead of i. He had the nasty habit of writing {%omega t - k x } instead of {k x - %omega t}.
DigitalMeatball wrote:([omega]*t - kx) isn't wrong. It just implies that your wave is propagating in the negative x direction, as opposed to (kx - [omega]*t), which propagates in the positive x direction. Inconsistency is another issue.
grim4593 wrote:Oye, we are in the middle of learning complex numbers in my Advanced Engineering Math class. We have covered all the complex functions and derivatives and on Monday are going to start integrals of complex numbers.
spatulated wrote:you know whats fun i^(-i) is a real number too! its like 1.3 something.
moopanda wrote:Dammit. I read this and thought "Ooh I'm a maths major I can contribute something meaningful here." but NOOO the topic seems just about covered. Then someone mentioned i, j and k and I thought hah I can be a smart arse about quaternions....but NOOOOO. So instead I'll just whinge about me being too slow
... I'm too slow.
Gelsamel wrote:/pat
You can explain to me how you get negative numbers through powers using i.
moopanda wrote:Of course we can get negative number through real powers as well (-1)^3 = -1
Gelsamel wrote:Doesn't that mean e^(i*x) = cis x?
Gelsamel wrote:Also would 5^a*i give me a negative number, ever?
Gelsamel wrote:So basically how the hell does e^(i*x) = cos x + i sinx?
aldimond wrote:spatulated wrote:you know whats fun i^(-i) is a real number too! its like 1.3 something.
I'm getting i^(-i) = e^(-i^2*pi/2) = e^(pi/2). Which is about 4.8.
i^i, of course, would be the reciprocal of that, so a bit more than a fifth. A bit more than a fifth, 'eh? Well, I don't make too much sense when I've had that much to drink either.
moopanda wrote:Sort of like a real number that is a multiple of 1 is an integer... but still a real number (and by extension, a complex number).
yy2bggggs wrote:Charon wrote:yy2bggggs wrote:i, j, k, -i, -j, -k... what's the difference?
Pretty important if you're talking about the quaternions
Any number whose square is -1 makes just as good an i as any other.
SpitValve wrote:Except when [i,j,k,l] are all different roots of -1... and ij=k, jk=l, li=j. And that's everything I know about quaternions.
ooh and you use them to rotate things.
SpitValve wrote:yy2bggggs wrote:Charon wrote:yy2bggggs wrote:i, j, k, -i, -j, -k... what's the difference?
Pretty important if you're talking about the quaternions
Any number whose square is -1 makes just as good an i as any other.
Except when [i,j,k,l] are all different roots of -1... and ij=k, jk=l, li=j. And that's everything I know about quaternions.
ooh and you use them to rotate things.
I'm amazed that nobody's mentioned noncommutativity of the quaternions here.
The Mighty Thesaurus wrote:Why? It does nothing to address dance music's core problem: the fact that it sucks.
no-genius wrote:we had to do i^i in Oscillations and Waves last year. Think it was e^ipi/2
Edit: no, it was e^pi/2. I just wasted a post
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