Well, I have nothing else to do, so I'm going to prove Euler's formula for all of you Newgrounder's benefit.
What's really interesting about Euler's formula is that it's so counter intuitive, that it is not uncommon for people to present it as proof of God's existence. It says that the natural constant, e, multiplied by itself x times the square root of negative times is equal to the square root of negative one times the ratio of a right triangle's opposite side to its hypotenuse if the triangle's angle is x, plus the ratio of a right triangle's adjacent side to its hypotenuse if the triangle's angle is x.
Or in math: e^(i * t) = i * sin(t) + cos(t)
Anyway, unto the proof:
i = sqrt( -1 ) and t be some variable:
y(t) = e^(i * t)
y'(t) = i * e^(i * t)
y''(t) = -1 * e^(i * t)
y(t) + y''(t) = 0
Taking the laplace transform of both sides:
L[ y(t) ] + L[ y''(t) ] = L[ 0 ]
y(s) + s * y'(s) - y'(0) = 0
y(s) + s^2 * y(s) - s * y(0) - y'(0) = 0
Alright, now going back to our equations
y(0) = 1
y'(0) = i
y''(0) = -1
Plugging those in we get
y(s) + s^2 * y(s) - s - i = 0
y(s) * (1 + s^2) - s - i = 0
y(s) = s / (1 + s^2) + i / (1 + s^2)
Taking the inverse Laplace transform of both sides (which is usually done through tables, since the transform itself is really ugly), we find that:
y(t) = cos(t) + i * sin(t)
y(t) = e^(i * t)
e^(i * t) = cos(t) + i * sin(t)
So there you have it, no magic, just maths.
Minkey2279
to... many... numbers... *starts twitching*