So if z=cosx+isinx, then dxdz=iz. If we assume separation of variables works(*) we go ∫z1dz=∫idx, so lnz=ix+C. Setting x = 0 we see C = 0 and so z=eix.
(*) If you don't want to assume this works, you can use the integrating factor idea from FM to realise that dxdze−ix is constant. Unless I'm missing something, the only unjustified step with this approach is the assumption that dxdeix=ieix.
So if z=cosx+isinx, then dxdz=iz. If we assume separation of variables works(*) we go ∫z1dz=∫idx, so lnz=ix+C. Setting x = 0 we see C = 0 and so z=eix.
(*) If you don't want to assume this works, you can use the integrating factor idea from FM to realise that dxdze−ix is constant. Unless I'm missing something, the only unjustified step with this approach is the assumption that dxdeix=ieix.
This is exactly how I first saw Euler's formula derived. Very interesting although I still haven't seen proofs/derivations of certain aspects of it like ∫z1=lnz etc.
This assumes that their is something special about cosx+isinx.
Well, yes, but there is something special about cos x + i sin x. It's a point on the unit circle. And "as any fule kno", when an object moves in a circle, its velocity (derivative) is at right angles (multiplied by +/- i) to the radius vector. In other words, the realization that cos x + i sin x has an "interesting" derivative is directly motivated by geometrical considerations.
In my proof (although I don't call it one ) you discover the result sponteneously, starting with the common pythagorean trig formula and deducing stuff, until your are led naturally to the result.
It's subjective, but I find my argument a lot more natural than yours. (Natural enough that I did actually derive this on the fly starting from thinking about complex numbers on rotation - although after about half a line i realised "I've seen this somewhere before").
Method 2: Let x=n+1 and y=n. Naturally squaring both equations we get x2=n+1 and y2=n, so eliminating n we obtain x2−y2=1, then (x−y)(x+y)=1, and finally x−y=x+y1. Back substituting for x,y we get n+1−n=1/(n+1+n), and then we continue as before.
This removes the rabbit element, essentially breaking down a "flash of insight" into easily understood pieces.
I don't think this is a great example: any experienced mathmo is instantly going to at least think about what happens when you multiply by the conjugate, and this leads you to method 1.
You might say the conjugate is an unbroken down "flash of insight", but if you break it down, the idea is much more "if x^2 and y^2 are nice and xy isn't, then (x-y)(x+y) is likely to be worth considering" than anything in your reasoning.