How to prove the following:$$|e^{-i\left
It looks like a beginning of Taylor series but I can't see how to get the inequality.
How to prove the following:$$|e^{-i\left
It looks like a beginning of Taylor series but I can't see how to get the inequality.
We actually have to prove that for a real number $t$, $|e^{it}-1|\leqslant |t|$. To see that, write $$e^{it}-1=\int_0^tie^{is}ds,$$ then use triangle inequality and the fact that $|ie^{is}|\leqslant 1$.
The inequality $$ |e^{i \varphi} - 1| \leqslant |\varphi| $$ simply states that the length of an arc in a unit circle is greater or equal to the length of the chord with the same endpoints. It is a well known fact )