3 ms·

The proof of the formula is beautiful. It's common to defined the complex exponential as the extension of the Taylor expansion of exp(x) to the complex plane. T

The proof of the formula is beautiful. It's common to defined the complex exponential as the extension of the Taylor expansion of exp(x) to the complex plane. Thus, exp(iy) = 1 + iy + (iy)^2/2! + ... Now just group the even-numbered terms together and group the odd-numbered terms together, the i's multiply to become 1 in the even numbered terms, and what you get is (the Taylor expansion of cos) + i(the Taylor expansion of sin)

escherplex 8y ago
Interesting. Looking at the Euler equation again as Argand plane rotation, would e^-ix be a form of clockwise rotation? Using the methodology of: https://www.mathsisfun.com/algebra/eulers-formula.html https://www.mathsisfun.com/algebra/eulers-formula.html as a reference template, e^-ix would seem to involve: (taylor cosine series) - i * (taylor sine series) or e^-ix = cos x - i sin x = -1 which suggests another twist to the familiar identity: e^ix * e^-ix = (cos x + i sin x) * (cos x - i sin x) = (cos x)^2 + (sin x)^2 = (-1) ^ 2 = 1 = e^0
compumike 8y ago
Indeed! I've written it out here with MathJax for equations: https://www.circuitlab.com/textbook/complex-numbers/ https://www.circuitlab.com/textbook/complex-numbers/ (Thanks to someone who found a small equation typo and emailed me!)