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Or an example from complex analysis: suppose you have a function that's analytic everywhere in the complex plane. Than integrate it round a closed curve in the complex plane, and you'll find that you get 0. Again, it's mindblowing!
For the last, you need to know what analytic means: (f(z+epsilon)-f(z))/epsilon approaches a constant limit as epsilon approaches 0, everywhere. Intuitively, the key thing going beyond real differentiability is that epsilon can approach 0 from any direction in the complex plane
Incidentally, do you know a nice proof of cauchys thm? The ones through Green felt like proof by calculating which is never very satisfying to me
One way: integrate f(z_0+delta) around a small rectangle, keeping terms to first order in delta. It's clearly 0 - the terms on the sides cancel, as do the terms at "top" & "bottom". Integration around any closed loop can be done thru telescopic cancellation of such small rects.
Really? I thought a function is analytic if it is infinitely differentiable and you are saying it is analytic if it is differentiable once. Maybe the latter implies the former. I never really took a complex analysis class.
The latter implies the former. It's quite remarkable.
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