Can someone point out what's wrong here? [Euler Equation]

ApeXaviour

Bored in a lecture, decided I'd just mess around with the euler equation, prove it right so to speak. Seems I couldn't. Now I saw in physics world where someone wrote in and explained that it ain't "mystical" and it can quite easily be solved by some expansion blah-dy-blah. I didn't really get what he was talking about tbh. ANYWAY, I was wondering either A) what am I doing wrong below? or If I'm not doing anything wrong, why the frick can it not be solved as below?

e^(pi) + 1 = 0
e^(pi) = -1
e^(2pi) = 1
2pi = ln(1)
2pi = 0
-4p^2 = 0 ???

Where p = Pi, I think the rest is obvious.

Eye

just a guess here, i could be waaaay off, and i probably am as it's been a few years since i did any physics/chemisty or anything like that.

e^(pi) = -1
e^(2pi) = 1

you changed the sign on the right side of the equation but did'nt on the left instead came up with 2Pi to balance the change from -1 to 1 .... something about that just dont seem right.

basically you multiplied the right side by -1 to get +1, should you do that same on the left?

maybe it is right and i'm way off (which i highly probable )

dudara

No, he squared it, not multiplied by -1.

exp(pi)^2 = exp(2pi)

as to the rest, I dunno. It's not my area of expertise

ecksor

Firstly, if you want to prove something then you must start with something you know and then work from there to get the conclusion you want, but this isn't the maths forum so we'll let you off with that.

Secondly, if you want to find the origins of the equation then you need to be familiar with complex numbers in a polar form and how to work out complex powers. Polar form being the modulus/argument form where you have c = r(cos(a) + i.sin(a)), r being the length of the line segment from the origin to c and a being the angle that that line segment makes with the real axis. If you've not seen that before then a bit of fiddling around with the trig definitions should convince you that it works out the appropriate lengths.

Also, if you know your power series, you know that e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...

If you extend this to allow complex powers (which we can do by definition since it doesn't contradict anything that we want to work nicely in the real numbers), then you have e^(ix) = 1 + ix + (i^2x^2)/2! + (i^3x^3)/3! + ...

Notice how i raised to even powers gives you real numbers flipping back and forth between negative and positive and by splitting up the real and imginary parts you can get:

e^(ix) = (1 - (x^2)/2! + (x^4)/4! - ...) + i(x - (x^3)/3! + (x^5)/5! - ...)

The above are the power series for cos and sin respectively, so we have that

e^(ix) = cos(x) + i.sin(x)

From there it's a simple matter to substitute pi for x and we have

e^(i.pi) = cos(pi) + i.sin(pi) = -1

If you want a more rigourous and detailed explanation of it then what you want is a copy of any introductory complex analysis text or possibly an physics/engineering methods book covering complex numbers. They generally cover it near the start.

ecksor

I decided to see if I knew any online notes that might be a good introduction. The pdf "Introduction to Complex Analysis for Engineers" by Michael D. Alder is free to download but the explanation of that is 75 pages in so mightn't be what you want. Then again, it might and is worth a look. Otherwise, "Complex Analysis" by John M. Howie has a decent explanation after 20 or so pages. However, in reference to the person who wrote in that it didn't have "mystical" properties, I'd have to wonder what they meant. We can derive the formula, but there's plenty of mystery there as Michael D. Alder writes after his derivation of it:

This links up the five most interesting numbers in Mathematics, 0; 1; e; i; pi,
in the most remarkable formula there is. Since e seems to be all about what
you get if you want a function f satisfying f' = f, and pi is all about circles,
it is decidedly mysterious.

Thinking about this gives you a creepy feeling up the back of the spine: it is
as though you went exploring the Mandelbrot set and found a picture of an
old bloke with a stern look and long white whiskers looking out at you. It
might incline you to be better behaved henceforth.

Capt'n Midnight

Saw on a web site, so it must be true, that you can rearragne parts of the equation to represent ways of looking at 180 degree angle.

This was not it..
http://www-structmed.cimr.cam.ac.uk/Course/Adv_diff1/Euler.html