In order to preserve the discussion of another thread, I thought I'd post this question in a new one.
What is a complex variable? the example I have is this: in the consideration that is a complex variable.

In order to preserve the discussion of another thread, I thought I'd post this question in a new one.
What is a complex variable? the example I have is this: in the consideration that is a complex variable.
I should have just said "variable". I just meant they can take on complex values.
Let's do the example n=1:
Here's an even more surprising example:
equals 0 for n odd, and equals 1 for n even.
and this is only true if for all i and j, if that's not already implied. For the case n=1, you have
would give and that is indeed 0. ), so one is the additive inverse(not sure on the proper term, but I know there is one) of the other and therefore sum to 0.
The case n=2 is the first n even, so we would have for that one: and this becomes: and this whole mess will become gigantic when expanded.
Yes, well, if the statement were trivial I wouldn't have posted it.
you see no beauty in some trivial concepts?
I'm not saying that.
When I first saw those kinds of formulas I was like "what? really?? That's crazy!" So wanted to share that feeling.
Ahh, okay, I agree with you there. Those two are rather elegant. And quite large for the second case, as at n=2 it already takes up the width of the paper I'm working on to demonstrate it true. I dunno, my first reaction on seeing that kind of problem is "Huuuuuh??" And then, later, once I get it I have the reverant "Holy Crap!!" sensation.
There are essentially a limitless number of formulas of that nature. Some of them are absolutely absurd, and quite beyond barehands verification. The most crazy formulas of this nature were conjectured by Witten using intuition from string theory.
The ones I have shared here are easy to prove if you look at them from the point of view of index theory (a theory relating the solutions of linear PDEs to topological invariants).
those two aren't horribly gross, just the initial start up was. And actually, the case for n=odd is a lot more obvious and intuitive than n=even. Gah, it provided me with a good 3 hours of fun though, thanks
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