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bio website natesoares.com
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age 25
visits member for 3 years, 8 months
seen Jul 24 at 3:26

Mar
14
awarded  Yearling
Mar
6
revised What's so special about sine? (Concerning $y'' = -y$)
corrected spelling, fixed formatting
Mar
6
suggested suggested edit on What's so special about sine? (Concerning $y'' = -y$)
Mar
3
comment What's so special about sine? (Concerning $y'' = -y$)
I understand that, but I don't completely buy this. Wherever there is a circle in nature, looking closer we find it not a circle. Orbits are actually ellipses, the earth isn't quite spherical, and so on.
Mar
3
comment What's so special about sine? (Concerning $y'' = -y$)
"Nearly every"? Which ones can't? Also, given the taylor series approximation for sine, can't anything approximated by a harmonic oscillator be modeled by even simpler functions?
Feb
29
comment What's so special about sine? (Concerning $y'' = -y$)
Thanks! I suppose I probably should take this question over to physics, I'm also curious as to why physics seems to care so much about the second derivative (in everything from F=ma to the position/momentum interaction).
Feb
29
accepted What's so special about sine? (Concerning $y'' = -y$)
Feb
29
awarded  Nice Question
Feb
28
awarded  Editor
Feb
28
revised What's so special about sine? (Concerning $y'' = -y$)
added 448 characters in body
Feb
28
asked What's so special about sine? (Concerning $y'' = -y$)
Jun
6
comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?
That helped a lot, thanks. In my own terms, the key point that I was missing was that taking $\lim_{n\to\infty}$ negates the drift off the unit circle that you see in $1+iX/n$, but does not negate the rotation caused by $1+iX/n$. It still astounds me that by taking the limit in this situation you negate the drift but not the rotation - that's an incredibly beautiful and powerful idea. :)
Jun
6
accepted Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?
Jun
6
comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?
I get it! Multiplying by $1+ix$ gives us rotation by x but doesn't keep us on the unit circle. However, if we divide by n and raise to the power n, we manage to quell the drift off the circle but maintain the rotation. The drift is lost if we divide by and raise to the power n, but the rotation is not. Looking deeper, is there any field of math concerned with which operations are susceptible to elimination by such limits and which operations are unaffected?
Jun
6
comment Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?
Thanks! I don't quite understand the image above: I understand the first frame represents $1 + i*\pi$, but I don't understand the following frames. I know I'm being a bit dense, but could you break frame 3 down for me into what each segment represents?
Jun
5
asked Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?
May
23
comment Complex Exponents
Thanks, that clears a lot of things up. I still don't quite understand why multiplication by (1+iX/N) is equivalent to rotation by X/N - do you think you could clear that up a bit for me?
May
18
comment Complex Exponents
See, that's exactly the part that I'm trying to understand by understanding complex exponents. Everybody just waves their hands and says "by definition" or pulls out a Taylor series, which is a proof, not a reason. I'm trying to figure out why $e^{ix}=cos(x) + i*sin(x)$. Lubos Motl's answer below helps, but if you have any extra pointers they'd be more than welcome.
May
18
accepted Complex Exponents
May
18
comment Complex Exponents
That definitely helped me understand why A^i = x+iy implies that (x,y) describes a point on the unit circle. However, it still doesn't help me understand how x and y are found, or why e plays such a big role. Can you give me some pointers to understanding those two things?