1. ## exponential

hi can someone please explain to me how u evaluate $e^{2 \pi i}$ And $e^{i \pi}$ .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?

2. Originally Posted by Dgphru
hi can someone please explain to me how u evaluate $e^{2 \pi i}$ And $e^{i \pi}$ .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
Let $\varphi$ be a real number, then $e^{i\cdot\varphi}$ is a point on the unit-circle (radius 1, center at 0) in the complex plane, where $\varphi$ 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 $e^{i\cdot\varphi}$ is $\cos(\varphi)$, and its imaginary part is $\sin(\varphi)$.
Overall, you get the relationship $e^{i\cdot\varphi}=\cos(\varphi)+i\cdot\sin(\varphi )$.

What you get, therefore, is in the case of $e^{2 \pi i}=\cos(2\pi)+i\cdot\sin(2\pi)=1+i\cdot 0=1$
and in the case of $e^{i\cdot \pi}=\cos(\pi)+i\cdot\sin(\pi)=-1+i\cdot 0=-1$.

3. Originally Posted by Dgphru
hi can someone please explain to me how u evaluate $e^{2 \pi i}$ And $e^{i \pi}$ .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
Euler's formula states that:
e^(ix) = cos x + i sin x

4. Originally Posted by Dgphru
hi can someone please explain to me how u evaluate $e^{2 \pi i}$ And $e^{i \pi}$ .. ? I remember my lecturer saying something about circles but I didn't understand. Please help me,?
I can't not answer this one

$e^{ix} = \cos(x) + i\sin(x)$

Therefore $e^{i\pi} = \cos \pi + i \sin \pi = -1$

By the same logic $e^{2\pi i} = \cos (2\pi) + i \sin (2\pi) = 1$

For any integer k: $e^{k\pi i} = \cos (k\pi) = \pm 1$ depending on if k is odd or even