Three proofs of Euler's formula: e^(it) = cos(t) + isin(t)

[phantom power]

I've wondered for years about the proof of this astonishing relation, which is a cornerstone for enormous bodies of mathematics and physics. For some reason, nobody I ever asked could point me to a proof. Once again, Wikipedia comes to my rescue with three nice proofs:

http://en.wikipedia.org/wiki/Euler%27s_formula#Proofs

Even after seeing it proven three ways, that equation still bakes my noodle. Where do I go to get my intuition back?

[Dudley_DUright]

1. I am teaching optics this semester and could not imagine

doing so without the handy-dandy Euler (pronounced "oiler" for those who have not seen it before) formula.

[BadgerKid]

2. Essentially it's a coordinate transformation

between polars and Cartesians. But I hadn't seen the diff eq. approach before. Thanks for the enlightenment. :thumbsup:

[mccoyn]

3. This formula floored me.

After a semester of multi-variable calculus I took differential equations. I just couldn't believe something could be so advanced, yet so eloquent. I swear for a week I thought I misunderstood it, but no, it really is that simple.

After that epiphany I took a few courses that should have had diff eq as a prerequisite, but were taught using clumsy concepts like phasors. They would have been much easier classes if they used Eulers formula and differential equations.

[Igel]

4. I remember seeing the Taylor's series

version of the proof in complex analysis when I took it at U. Delaware.

Fun course. I wish I could remember it.

And I certainly wish I had taken it before I had a disastrous time in circuit analysis class.

[phantom power]

5. I once took information theory before the graduate-stats prereq.

Worst semester of my life. I can only assume the prof wondered why I wasted so much paper proving results from graduate statistics. The only class I ever took where I got a grade based on "effort." Ugh.

[hunter]

6. You build an entire system of mathematics upon these properties...

... and then you say, "Look at that! These are the properties of my mathematics!"

There's a kind of circular reasoning in any proof like this. Sometimes it condenses down to something very small.



The wonder of this may be entirely a wonder of our minds and and not necessarily a wonder of this universe (of which our minds are a part -- oh no, more circles to chase!)

One appropriate response, rather than seeking proofs, is "Let's go drink some beer."

:beer:

[bemildred]

7. It is more a definition than a proof.

The "proof" is a justification of the definition, or a demonstration that it must be that way. The Complex plane and Real line are unique up to isomorphism, and so is the method to embed the latter in the former (I think).
:beer: