I once had an interesting thought about the function e^x. I think this is a key idea in the theory of Lie groups.
If the x in e^x = (1+x/N)^N is understood as some transformation, then e^x is essentially repeating an infinitesimal transformation lots of times. So it's like a for-loop where the body of the loop is some infinitesimal transformation.
I tried to define the integration operator in terms of e^x. The 1 + x/N needed to be one "infinitesimal" iteration of integration, that adds an extra infinitesimal rectangle to the area. But it didn't seem to work out.
Ultimately that helps to explain Euler's formula. For large N, 1+ix/N is an "infinitesimal" rotation by angle x/N. Repeating it N times produces a rotation of angle x. That's essentially what TFA says. And it's a special case of the Lie theoretic view of e^x.
If the x in e^x = (1+x/N)^N is understood as some transformation, then e^x is essentially repeating an infinitesimal transformation lots of times. So it's like a for-loop where the body of the loop is some infinitesimal transformation.
I tried to define the integration operator in terms of e^x. The 1 + x/N needed to be one "infinitesimal" iteration of integration, that adds an extra infinitesimal rectangle to the area. But it didn't seem to work out.
Ultimately that helps to explain Euler's formula. For large N, 1+ix/N is an "infinitesimal" rotation by angle x/N. Repeating it N times produces a rotation of angle x. That's essentially what TFA says. And it's a special case of the Lie theoretic view of e^x.