4 ms·
is there anything interesting to you in the notion that cos(x)+isin(x) is the unique analytic continuation of e^x in the complex plane? It's not just a rotatio
by ouid 9y ago
is there anything interesting to you in the notion that cos(x)+isin(x) is the unique analytic continuation of e^x in the complex plane?
It's not just a rotation by definition, I don't think.
- deleted 9y ago[deleted]
- jacobolus 9y agoIf you think about it for a bit, it’s clear that this must be so. A circle’s circumference is proportional to its radius, so if you want to have a conformal map between a rectangular grid in one space and a polar-coordinate grid in another space, the spacing of the polar-coordinate grid lines in the radial direction must necessarily be proportional to the radius so that they can match the radius-dependent spacing in the tangential direction. Anyhow, there are many interesting features of the structures here. Both rotation and logarithms are very interesting and important both in mathematics and in science/engineering. I just don’t think Euler’s identity in particular is worthy of so much attention.
- ouid 9y agoSure, but I think most people believe that cos(x)+isin(x)=e^ix because that's how it is defined, in which case e^(i\pi)=-1 isn't even a result. It is at least the tiniest bit more interesting than that.