hi can someone please explain to me how u evaluate $\displaystyle e^{2 \pi i} $ And $\displaystyle e^{i \pi} $ .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
Let $\displaystyle \varphi$ be a real number, then $\displaystyle e^{i\cdot\varphi}$ is a point on the unit-circle (radius 1, center at 0) in the complex plane, where $\displaystyle \varphi$ is the angle in radians corresponding to that point as seen from the origin 0, and relative to the positive direction of the real axis.
Because of this, the real part of that complex number $\displaystyle e^{i\cdot\varphi}$ is $\displaystyle \cos(\varphi)$, and its imaginary part is $\displaystyle \sin(\varphi)$.
Overall, you get the relationship $\displaystyle e^{i\cdot\varphi}=\cos(\varphi)+i\cdot\sin(\varphi )$.
(See also: Euler's formula - Wikipedia, the free encyclopedia).
What you get, therefore, is in the case of $\displaystyle e^{2 \pi i}=\cos(2\pi)+i\cdot\sin(2\pi)=1+i\cdot 0=1$
and in the case of $\displaystyle e^{i\cdot \pi}=\cos(\pi)+i\cdot\sin(\pi)=-1+i\cdot 0=-1$.
I can't not answer this one
$\displaystyle e^{ix} = \cos(x) + i\sin(x)$
Therefore $\displaystyle e^{i\pi} = \cos \pi + i \sin \pi = -1$
By the same logic $\displaystyle e^{2\pi i} = \cos (2\pi) + i \sin (2\pi) = 1$
For any integer k: $\displaystyle e^{k\pi i} = \cos (k\pi) = \pm 1$ depending on if k is odd or even