/sci/ - Science & Math

>>9744512
>I’ll have those high schoolers wanting to major in Pure Math for the next four hundred years or so

>>9744512
Why do people hold this identity with such high regard? It's pretty useless.

e^ix = cosx + isinx is much more useful and beautiful

>>9744961
It's because for the uninitiated it looks mysterious and awe-inspiring. It combines 5 important numbers in a succint equation.
After learning basic complex numbers though it does lose its charm a bit.

I still don't get why f(z) = ze^{iθ} rotates a point z by θ degrees

>>9745195
Because by Euler's identity, e^(ix) = cos(x) + i*sin(x), (use the power series of e^z + basic properties of analytic functions to prove this). If it's the elementary trigonometry that fucks you over, then you will never make it.

>>9744961
>>9745182

Also, there is a problem with people trying to look better and smarter than other people but, at the end of the day, the essentially wrote 2+2=4 on their goddamn arms and called themselves a genius. This is what the cancer of pop sci does, it makes people feel smart but lazily writing down the most basic of the basic shit in the field. I wouldn't hate this equation so much if it wasn't tattooed on every brainlet that every tried to pass complex analysis.

>>9745195
Because it preserves the modulus, |f(z)|=|z||e^iθ|=|z| since e^iθ = cosθ + isinθ is a point on the unit circle. And it changes the angle by exactly θ (mod 2pi), arg(f(z))=arg(z)+arg(e^iθ)=arg(z)+θ.

>>9744961
>e^ix = cosx + isinx
I thought I found a unique way to derive this in high school but now I can't remember what I did. It was about three pages handwritten. Most likely, I just used postulates from my trig textbook that actually relied on the identity and I didn't know it.

>>9745701
The proof itself is fairly simple once you establish that exp(z) is analytic in C. Pic related.

>>9745701
>>9745735
https://59clc.files.wordpress.com/2011/01/real-and-complex-analysis.pdf
Just define e^z as a series then use that to define trig functions