/sci/ - Science & Math

>>16131221
yee haw

>>15925528
it makes sense to me as the pythagorean formula applied to a right triangle with 1 real leg longer than the real hypotenuse, forcing the other leg to go imaginary. I was able to get Euler's formula by farting around with the single-variable function and parametric equation ways of expressing a circle and out-of-bounds hyperbola or vice-versa, getting some sorta expression with arcsinh(arcsin()), then differentiating and immediately integrating. That last part was a simple enough way to get around sin and arcsin not having closed forms, but I haven't seen it done before and I'm not sure if it's correct or if it just agrees with the answer I was looking for. I wrote it all down but I'm not at home, might have time tomorrow to rewrite it and post. All in all I don't find that 'trick' much more enlightening than the usual proof through infinite series.

I'm pretty sure the root of the connection is when you force the Pythagorean identity on improper right triangles, on right triangles where x > r, so that y has to be complex in order for the x^2 + y^2 = r^2 to hold. I'm looking for a clean way to show it with geometry/algebra, not sure if it's possible